Lucas' theorem modulo $p^2$

Lucas’ theorem describes how to reduce a binomial coefficient $\left(\genfrac{}{}{0px}{}{a}{b}\right)$ modulo $p$ by breaking off the least significant digits of $a$ and $b$ in base $p$. We characterize the pairs of these digits for which Lucas’ theorem holds modulo $p^2$. This characterization is naturally expressed using symmetries of Pascal’s triangle.