  Protons and neutrons in the atomic nucleus move in shells analogous to the
electronic shell structures of atoms. Nuclear shell structure varies across the
nuclear landscape due to changes of the nuclear mean field with the number of
neutrons $N$ and protons $Z$. These variations can be probed with mass
differences. The $N=Z=40$ self-conjugate nucleus $^{80}$Zr is of particular
interest as its proton and neutron shell structures are expected to be very
similar, and its ground state is highly deformed. In this work, we provide
evidence for the existence of a deformed double shell closure in $^{80}$Zr
through high precision Penning trap mass measurements of $^{80-83}$Zr. Our new
mass values show that $^{80}$Zr is significantly lighter, and thus more bound
than previously determined. This can be attributed to the deformed shell
closure at $N=Z=40$ and the large Wigner energy. Our statistical Bayesian model
mixing analysis employing several global nuclear mass models demonstrates
difficulties with reproducing the observed mass anomaly using current theory.
