  In this paper, we consider domino tilings of regions of the form $\mathcal{D}
\times [0,n]$, where $\mathcal{D}$ is a simply connected planar region and $n
\in \mathbb{N}$. It turns out that, in nontrivial examples, the set of such
tilings is not connected by flips, i.e., the local move performed by removing
two adjacent dominoes and placing them back in another position. We define an
algebraic invariant, the twist, which partially characterizes the connected
components by flips of the space of tilings of such a region. Another local
move, the trit, consists of removing three adjacent dominoes, no two of them
parallel, and placing them back in the only other possible position: performing
a trit alters the twist by $\pm 1$. We give a simple combinatorial formula for
the twist, as well as an interpretation via knot theory. We prove several
results about the twist, such as the fact that it is an integer and that it has
additive properties for suitable decompositions of a region.
