  We study the coordinate ring of an $L$-convex polyomino, determine its
regularity in terms of the maximal number of rooks that can be placed in the
polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those
which are Gorenstein on the punctured spectrum, and compute the Cohen--Macaulay
type of any $L$-convex polyomino in terms of the maximal rectangles covering
it.
