Half Turn Cube is a Rubik's Cube that only allows half turns. This makes it a different puzzle that is easier to solve. It was contrived by Takafumi Haseda, and first prototyped by Hidetoshi Takeji in 2007. Read about that prototype at the Twisty Puzzles Forum.
Yes, a half-turn 2x2x2 is possible. As you may know, most 2x2x2 on the market have a smaller 3x3x3 inside, of which you can only see the corners. If you modify the Half Turn Cube into a 2x2x2 you get the desired effect.
Someone else suggested a more fiendish variation: build a 2x2x2 with a bandaged 3x3x3 inside. This way your 2x2x2 sometimes turns and sometimes it blocks, but you cannot see why it blocks.