  We establish a version of Kn\"{o}rrer's Periodicity Theorem in the context of
noncommutative invariant theory. Namely, let $A$ be a left noetherian
AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive
degree, and take $B=A/(f)$. Then there exists a bijection between the set of
isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules
over $B$ and those over (a noncommutative analog of) its second double branched
cover $(B^\#)^\#$. Our results use and extend the study of twisted matrix
factorizations, which was introduced by the first three authors with Cassidy.
These results are applied to the noncommutative Kleinian singularities studied
by the second and fourth authors with Chan and Zhang.
