  Twisted generalized Weyl algebras (TGWAs) are a large family of algebras that
includes several algebras of interest for ring theory and representation
theory, such as Weyl algebras, primitive quotients of $U(\mathfrak{sl}_2)$, and
multiparameter quantized Weyl algebras. In this work, we study invariants of
TGWAs under diagonal automorphisms. Under certain conditions, we are able to
show that the fixed ring of a TGWA by such an automorphism is again a TGWA. In
particular, this is true for $\Bbbk$-finitistic TGWAs of type $(A_1)^n$ and
$A_2$. We apply this theorem to study properties of the fixed ring, such as the
noetherian property and simplicity. We also look at the behavior of simple
weight modules for TGWAs when restricted to the action of the fixed ring. As an
auxiliary result, in order to study invariants of tensor products of TGWAs, we
prove that the class of regular, $\mu$-consistent TGWAs is closed under tensor
products.
