  We consider weighted shift operators having the property of moment infinite
divisibility; that is, for any $p > 0$, the shift is subnormal when every
weight (equivalently, every moment) is raised to the $p$-th power. By
reconsidering sequence conditions for the weights or moments of the shift, we
obtain a new characterization for such shifts, and we prove that such shifts
are, under mild conditions, robust under a variety of operations and also rigid
in certain senses. In particular, a weighted shift whose weight sequence has a
limit is moment infinitely divisible if and only if its Aluthge transform is.
We also consider back-step extensions, subshifts, and completions.
