My feeling: Use dragging, as in drag and drop. You can make it more complex, the pieces may come out from different "holes" in the wall, not just down, so you have to build things in a way that you have passages to reach from each hole.
If you try the game on mobile that's actually how the controls work. Although you need to be able to rotate before placing so simple dragging and dropping wouldn't work...
Unless I make the player hold the left mouse button while hitting right to rotate... Could work?!
You drag and drop, as usual, if another "touch" is detected, you rotate. The problem in the mobile is that the finger will "cover" the piece. So you have to drag and draw the piece with some "offset from the finger".
