There is a program called ds4windows that makes playstation controllers look like xbox controllers for the operating system. ds4windows lets you assign buttons freely, and also allows to assign analog stick directions to other directions.

Here is how to invert the x-axis.

0. Disconnect your dualshock controller
1. Download ds4windows http://ds4windows.com/
2. Run it
3. Connect your dualshock controller with a usb cable
4. Create a new profile
5. In the "Controls" tab, assign "RS Left" to "Right X-Axis+" and "RS Right" to "Right X-Axis-"
6. Save the profile with a name like "inverted" and make sure that the dualshock uses this profile