  Twisted generalized Weyl algebras (TGWAs) $A(R,\sigma,t)$ are defined over a
base ring $R$ by parameters $\sigma$ and $t$, where $\sigma$ is an $n$-tuple of
automorphisms, and $t$ is an $n$-tuple of elements in the center of $R$. We
show that, for fixed $R$ and $\sigma$, there is a natural algebra map
$A(R,\sigma,tt')\to A(R,\sigma,t)\otimes_R A(R,\sigma,t')$. This gives a tensor
product operation on modules, inducing a ring structure on the direct sum (over
all $t$) of the Grothendieck groups of the categories of weight modules for
$A(R,\sigma,t)$. We give presentations of these Grothendieck rings for $n=1,2$,
when $R=\mathbb{C}[z]$. As a consequence, for $n=1$, any indecomposable module
for a TGWA can be written as a tensor product of indecomposable modules over
the usual Weyl algebra. In particular, any finite-dimensional simple module
over $\mathfrak{sl}_2$ is a tensor product of two Weyl algebra modules.
