The Twiddle shuffling algorithm was theoretically parity-unbalanced:

The Twiddle shuffling algorithm was theoretically parity-unbalanced:
it performed a fixed number of shuffling moves, and on each one it
had a 2/3 chance of flipping the permutation parity and a 1/3 chance
of keeping it the same. Markov analysis shows that over a run of
1500-odd shuffle moves this will end up being an undetectably small
actual bias in the parity of the generated grid, but it offends my
sense of pedantry nonetheless so here's a small change to make the
number of shuffling moves itself have randomly chosen parity. The
parity of generated grids should now be _exactly_ 50:50.

[originally from svn r5742]