Let $A$ be the first Weyl algebra, in the Euler gradation. We classify graded rings $B$ such that $gr-A$ and $gr-B$ are equivalent (we say that $A$ and $B$ are graded equivalent), and produce some surprising examples. In particular, we show that $A$ is graded equivalent to an idealizer in a localization of $A$. In the process, we derive a concise new characterization of equivalences of graded module categories that generalizes the classical Morita theorems.