In Duet, you help two vessels navigate through many courses of obstacles. Vessels are rigidly connected and have a pivot point in the center of the circle. You must rotate the circle with vessels to avoid them hitting white walls. If you hit a wall, it becomes painted with splash of vessel's respective color and you are returned to the beginning of the level.
I made a list of gamemodes and obstacles you'll encounter during your playthrough, because I quite like this game and no one made it yet. I hope the hints I mentioned here will help some of the readers.

Story is a first gamemode in the game. It teaches you gameplay basics and introduces you to obstacles. Also it has very good narration and music.
It has 9 chapters, each one introducing some new obstacle or gameplay feature to the game. After completing a level, you can try to "perfect" it - complete it with least amount of moves necessary. This can be quite difficult and sometimes requires very accurate positioning of the vessels. However, the only your reward will be green triangle at the corner of level selection screen.
Completing each chapter gives you a corresponding achievement. Another achievement is earned for completing each chapter without deaths, which can be difficult in later levels.

In the first few levels narrator tells us how to navigate the course and gives several tips. After that you are on your own, and must learn how to avoid new obstacles by yourself.

• ignorance
  Basic blocks are introduced in this chapter.
  Beating the first level gives you "a circular state of mind" achievement.
  1. Plank
  
  A long rectangle, which appears on right or left. Easily avoided by rotating.
  Several planks in a row can be avoided without releasing the left/right button, resulting in a combo, which is explained in ignorance-III level.
  
  2. Center plank
  
  A small plank which is avoided by placing the vessels in the initial horizontal position. It may not break the plank combo if you began it in the right moment.
  3. Pair of boxes
  
  The two big squares, which come from different sides. They can be avoided in the same way as planks, but if you are accurate, you can place vessels in vertical position, and you'll slip through the boxes without much effort. The first box also can be part of plank combo, if it follows planks from the same side.
  
• denial
  4. Rotating plank
  
  This plank is avoided by sychronising your movement with plank rotation and doing half-circle around it. Easy way to know what side you must move to is to look at the top point of the circle drawn by plank's ends. If they move to the right in the top point then you should push the right button, if they move to the left - then push left.
  Navigating around one of those gives you "spin class" achievement.
  5. Two planks.
  
  A normal plank, followed by a small plank on the other side. At first, you should rotate around it like around normal plank, but stop halfway, so the second vessel can pass through the gap.
  
• anger
  Here, some planks and boxes are "shifted" to 1 position backwards, so two blocks appear together from the top, but quickly move to their normal position. Easily avoided, if you can quickly react on changing environment.
  
  
• bargaining
  6. Big rotating plank
  
  They come from right or left and, when coming near vessels, they are located in horizontal position, what makes it similar to normal basic planks. Avoided in the same way as normal planks.
  
• guilt
  As if "shifting" was not enough, developers made it that blocks move with different speed until they get to their normal position. Actually, it's not really harder than "shifting".
  
  
• depression
  To confuse you even more, now some planks and boxes are constantly moving from right to left.
  Sometimes they are arranged in a way, that the plank combo is possible with them. Look for incoming planks: if two planks which follow one another have nearly synchronised movement from right to left - that's where the combo ends and you must move in the other direction.
  
  
• hope
  Didn't see any changes here, except sudden speed increase.
  
• acceptance
  Now the planks disappear when they come near you, so you should memorize their order while they are still visible. However, if you paint them - it doesn't disappear (yet), so it can help you navigating in the darkness.
  
  Navigating around an invisible plank gives you "invisible blocks!" achievement.
  
  Completing this chapter and watching credits to the end gives you "appreciation" achievement.

These are bonus levels and chapters which are meant to test your reflexes, endurance and memory even more. Each challenge is unlocked after completing a certain storymode chapter, so you can't see the new wall types before being introduced to them in story modes.
Here is a list of what you can encounter here.

• endless
  This is basically a score challenge. You are presented with a row of levels, next is harder than the previous and gets new blocks in the same order they appear in story mode. The score is the distance you have travelled in the course. You have 3 lives, if you die, a life will refill after you get 100 points.
  You can get three "infinite milestone" achievements here.
  
• daily
  Those are sets of 3 levels which change every day. Their difficulty depends on the day of the week: on Monday and Thursday you have easiest levels, on Tuesday and Friday you get medium levels, on Wednesday and Saturday you get hard levels and on Sunday you get very hard levels.
  After completing certain amount of challenges, you will get "beside myself", "ten times tables" and "centurion" achievements, and if you complete challenges in a row, you will get "triple-play" and "coming up sevens" achievements. Luckily, you can complete dailies on one day just by changing the system time and restarting the game.
  
• endless epilogue
  Another score challenge, now featuring obstacles added by Epilogue gamemode. Rules are similar.
  
• nightly challenges
  The same thing as daily challenges, but features obstacles added by Epilogue gamemode. Also new challenge is distributed not at the midnight, but at the midday.
  After completing 5 of those challenges you will get "royal flush" achievement.
  
• endless encore
  The last score challenge featuring obstacles added by Encore gamemode. Rules are also similar.

The last four challenges are extra chapters without any narration, but with extra hard themed levels. They also have achievements for completing them and for completing them without deaths. However, these levels are randomised every time you restart them, so you can't remember level layout, but should improvise instead.

• revolution
  This chapter is translated as "rotation" in some languages and that is exactly what it is about. revolution-I has 7 big rotating planks, revolution-II has 7 normal rotating planks, and revolution-III has 12 of both types, randomly distributed.
  
• quickening
  This chapter is all about speed and you won't encounter anything here except normal planks and centered planks. As speed increases with each level, you should use centered planks and "perfection" level endings for relaxation. quickening-V and quickening-VI don't have any centered planks, so you shouldn't really worry about breaking your plank combos. Still they are hardest levels in whole chapter.
  
• resilience
  This chapter has everything. It has rotating and moving blocks, shifting blocks and resilience-III even has disappearing blocks.
  
• transcendence
  This chapter has only basic blocks: planks, centered planks and boxes, but all of them quickly become invisible after appearing on the top of the screen. Death paint lasts a bit longer than blocks, but quickly disappears too, so you should trust your memory and intuition.
  After completing this chapter for second time I got extra-hard transcendence-IV level which was a hybrid of transcendence-III and quickening-VI.

This is another story and there are 7 chapters. Most of the levels are situated on cylindrical surface and blocks don't move straight down here, but at some angle. When blocks reach the edge of cylinder, they continue moving from the other side. This can be quite confusing at first. Also, some new block types are introduced.
Again, each chapter has a completion achievement and a no-death achievement.

• trust
  The introduction chapter. Cylindrical surface is first shown here and you get used to strangely moving blocks.
  
  
• understanding
  7. Arrowed planks
  
  There are two types of them, as seen on the picture. The first one behaves like normal plank at first, but divides in two after first vessel got around it, so you must stop rotating at nearly verticall position. The second is inversed version of the first one - you begin at vertical position and then rotate around plank.
  There is a way to bypass several of those planks in a row without moving by accurate positioning of the vessels, but it is quite hard and it's only used for perfecting levels.
  
• control
  In this chapter, rotating planks return. And now they don't move in straight line, what makes them a bit more dangerous, especially around other blocks.
  
• identity
  Here, blocks are quickly moving from right to left, so you can't predict at what position they will stop. It's not quite hard actually, if you had completed story mode before.
  
  
• initiative
  "Shifted" and different speed blocks return in this chapter. It's not really hard too.
  
• intimacy
  8. Pairs of squares
  
  This new block type is very dangerous and very hard to predict. It has two states: 1) both squares located in center - then you must place your vessels horizontally, 2) squares are located on the sides - then you place vessels vertically. If several of these moving squares are going in a row - you can predict how will behave squares after first one by looking at their synchronisation. But for single randomly located squares in later levels you will have to improvise and remember their state.
  
• integrity
  Just a compilation of all the previous obstacles. Also side-to-side moving blocks are encountered here (I'm not sure if they weren't encountered before). As all the blocks not moving vertically down, that makes you feel that those moving blocks follow quite a weird pattern.

Third part of the story in five chapters. Now blocks are not just going from the top, but they are going from the bottom of the screen too. This feels strange at first, but you get used to it in several tries.
Each chapter has it's own pair of achievements again.

• anticipation
  Introduction chapter with basic blocks. You should get used to the new gameplay with it.
  
  
• ambition
  Two new types of planks is introduced in this chapter. They resemble each other but behave quite different, so you must look and move carefully if you don't want to die.
  9. Direction-changing planks
  These planks are arrow-shaped but the difference between the types is in direction they are pointing to.
  Here, the arrowhead is facing the center of the screen, so when it gets to the center it will rotate and go backwards.
  
  Second type has arrowhead facing the screen edge, and it will rotate differently. You should treat this plank as normal plank at first, but when it starts moving, you should change direction of rotation, so you won't collide. It will look like the plank is chasing you.
  
  
• tension
  Arrowed planks return here. As they are going from top and bottom now, the situation becomes even more confusing and dangerous. The trick which is allowing to bypass several of them still works, but requires accuracy again.
  
  
• perspective
  10. Vertical and horizontal pair
  This chapter has something resembling pair of planks from the chapter 2 of story mode. However, these planks are always going from different directions. Still, it shouldn't be a problem, just don't make full circle around the horizontal planks.
  
• release
  Levels in this chapter have all the new blocks from previous chapters and also there are pairs of synchronised direction-changing plank (they appear at the same time and become one single long rotating plank for a second). The levels are a bit harder, but nothing impossible at all.

This is a time challenge. Here you can beat each of the three story modes and should do it as quick as you can. Also you can change your gamespeed, from normal 100% to 200%. After you complete it, you will be added to global leaderboards.


I hope that my guide will help understand some gameplay features to people, who feel that the game is quite hard for them.
Please make suggestions how to improve this and report mistakes in comments.