  We consider a class of models describing the dynamics of $N$ Boolean
variables, where the time evolution of each depends on the values of $K$ of the
other variables. Previous work has considered models with dissipative dynamics.
Here we consider time-reversible models, which necessarily have the property
that every possible point in the state-space is an element of one and only one
cycle.
  As in the dissipative case, when K is large, typical orbit lengths grow
exponentially with N, whereas for small enough K, typical orbit lengths grow
much more slowly with N. The numerical data are consistent with the existence
of a phase transition at which the average orbit length grows as a power of N
at a value of K between 1.4 and 1.7. However, in the reversible models the
interplay between the discrete symmetry and quenched randomness can lead to
enormous fluctuations of orbit lengths and other interesting features that are
unique to the reversible case.
  The orbits can be classified by their behavior under time reversal. The
orbits that transform into themselves under time reversal have properties quite
different from those that do not; in particular, a significant fraction of
latter-type orbits have lengths enormously longer than orbits that are time
reversal-symmetric. For large K and moderate N, the vast majority of points in
the state-space are on one of the time reversal singlet orbits, and a random
hopping model gives an accurate description of orbit lengths. However, for any
finite K, the random hopping approximation fails qualitatively when N is large
enough ($N\gg 2^{(2^K)}$).
