  The number of domino tilings of a region with reflective symmetry across a
line is combinatorially shown to depend on the number of domino tilings of
particular subregions, modulo 4. This expands upon previous congruency results
for domino tilings, modulo 2, and leads to a variety of corollaries, including
that the number of domino tilings of a k x 2k rectangle is congruent to 1 mod
4.
