  We define two classes of colorings that have odd or even chirality on
hexagonal lattices. This parity is an invariant in the dynamics of all loops,
and explains why standard Monte-Carlo algorithms are nonergodic. We argue that
adding the motion of "stranded" loops allows for parity changes. By
implementing this algorithm, we show that the even and odd classes have the
same entropy. In general, they do not have the same number of states, except
for the special geometry of long strips, where a Z$_2$ symmetry between even
and odd states occurs in the thermodynamic limit.
