  We consider domino tilings of three-dimensional cubiculated regions. A flip
is a local move: two neighboring parallel dominoes are removed and placed back
in a different position. The twist is an integer associated to each tiling,
which is invariant under flips. A balanced quadriculated disk $D$ is regular if
whenever two tilings $t_0$ and $t_1$ of $D \times [0,N]$ have the same twist
then $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra
vertical space is allowed. We define the domino group of a quadriculated disk
and prove that $D$ is regular if and only if its domino group is isomorphic to
$Z \oplus Z/(2)$. We prove that a rectangle $D = [0,L] \times [0,M]$ with $LM$
even is regular if and only if $\min\{L,M\} \ge 3$ and conjecture that in
general "large" disks are regular. In the cases where $D$ is not regular we
prove partial results concerning the structure of the domino group: the group
is not abelian and has exponential growth. We also prove that if $D$ is regular
then the extra vertical space necessary to join by flips two tilings of $D
\times [0,N]$ with the same twist depends only on $D$, not on the height $N$.
