  We consider domino tilings of three-dimensional cubiculated manifolds with or
without boundary, including subsets of Euclidean space and three-dimensional
tori. In particular, we are interested in the connected components of the space
of tilings of such regions under local moves. Building on the work of the third
and fourth authors, we allow two possible local moves, the flip and trit. These
moves are considered with respect to two topological invariants, the twist and
flux.
  Our main result proves that, up to refinement: Two tilings are connected by
flips and trits if and only if they have the same flux. Two tilings are
connected by flips alone if and only if they have the same flux and twist.
