Here is a list of all rotations systems known to be in use by various Tetris implementations: http://harddrop.com/wiki/Category:Rotation_Systems

By rotating the piece twice (and never reifying the intermediate position) immediately before/during the step that moves the pieces down

It's not so easy. You also need a small overhang above the hole, and make the rotation "kick" against the overhang to make the piece fit. I think in some implementations the rules for wall kicks actually allow you to climb up certain shaped walls, all the way back to the top of the screen.

I guess you are thinking of other orientations than straight up, as it is trivial to clear three lines with a T-piece:

   .....TTT..
   ......T...
   ......T...
   ..........

That's not a T piece. It has 5 blocks. The subtlety is about a board like

    .XXXXXXXXX
    ..XXXXXXXX
    .XXXXXXXXX

and if you can put a T piece inside of the gap on the left.

Oops. My mind is more pentomino-oriented, I guess.

All seven pieces ("tetrads", from "tetromino") in standard tetris have 4 blocks, not 5.