Conformations of Cyclohexane
Short answer, no. There is not a "twist-chair."

To convince yourself, it probably helps to make a physical model with a chemistry model kit. If you try to twist the chair, you can't do it without significantly moving at least one atom, in which case, the conformation is basically a "skew" or "twist boat." Try it.

The boat and chair conformations are indeed symmetric and achiral. The twist-boat or skew conformation has point group $D_{2}$ and is indeed chiral.

A normal half-chair will have an axis of symmetry and is chiral.

An envelope conformation will have a plane of symmetry and thus be achiral.

I've found that some of the best depictions of six-member-ring conformations come from pyranose sugars. Because of the hydroxyl groups and oxygen atoms, there are 38 distinct conformations (2 chairs, 6 boats, 6 skew-boats, 12 half-chairs, and 12 envelopes).

From Wikipedia:


  


Obviously, in less-substituted cyclohexane, there are fewer unique shapes, but it helps to see the different ring-puckering or pseudo-rotation shapes.