  The maximal cut of the nonplanar crossed box diagram with all massive
internal propagators was long ago shown to encode a hyperelliptic curve of
genus 3 in momentum space. Surprisingly, in Baikov representation, the maximal
cut of this diagram only gives rise to a hyperelliptic curve of genus 2. To
show that these two representations are in agreement, we identify a hidden
involution symmetry that is satisfied by the genus 3 curve, which allows it to
be algebraically mapped to the curve of genus 2. We then argue that this is
just the first example of a general mechanism by means of which hyperelliptic
curves in Feynman integrals can drop from genus $g$ to $\lceil g/2 \rceil$ or
$\lfloor g/2 \rfloor$, which can be checked for algorithmically. We use this
algorithm to find further instances of genus drop in Feynman integrals.
