  We present the first results of a deep $Chandra$ observation of the galaxy
cluster RBS 797, whose previous X-ray studies revealed two pronounced X-ray
cavities in the east-west (E-W) direction. Follow-up VLA radio observations of
the central active galactic nucleus (AGN) uncovered different jet and lobe
orientations, with radio lobes filling the E-W cavities and perpendicular jets
showing emission in the north-south (N-S) direction over the same scale
($\approx$30 kpc). With the new $\sim$427 ks total exposure, we report the
detection of two additional, symmetric X-ray cavities in the N-S direction at
nearly the same radial distance as the E-W ones. The newly discovered N-S
cavities are associated with the radio emission detected at 1.4 GHz and 4.8 GHz
in archival VLA data, making RBS 797 the first galaxy cluster found to have
four equidistant, centrally-symmetric, radio-filled cavities. We derive the
dynamical and radiative ages of the four cavities from X-ray and radio data,
respectively, finding that the two outbursts are approximately coeval, with an
age difference of $\lessapprox$10 Myr between the E-W and N-S cavities. We
discuss two scenarios for the origin of the two perpendicular, equidistant
cavity systems: either the presence of a binary AGN which is excavating coeval
pairs of cavities in perpendicular directions, or a fast ($<$10 Myr) jet
reorientation event which produced subsequent, misaligned outbursts.
