Update:
I think it's a glitch. The back buttons are supposed to echo the mouse buttons but your tools just disappear.
While we are speaking of glitches, can we have hands in the intro and a way of skipping the bloke's first lines or movement?
Cheers.

The 360 controls should be as follows.

L3 to run, tap A to sprint, Y to jump, B to crouch,
LT to use left item, RT to use right item, LB to take/drop left item, RB to take/drop right item.

And yeah, we're looking into why the hands aren't showing in the intro ^^; it's kinda wonky.