A leading explanation for the origin of Galactic cosmic rays is acceleration at high-Mach number shock waves in the collisionless plasma surrounding young supernova remnants. Evidence for this is provided by multi-wavelength non-thermal emission thought to be associated with ultrarelativistic electrons at these shocks. However, the dependence of the electron acceleration process on the orientation of the upstream magnetic field with respect to the local normal to the shock front (quasi-parallel/quasi-perpendicular) is debated. Cassini spacecraft observations at Saturn's bow shock have revealed examples of electron acceleration under quasi-perpendicular conditions, and the first in situ evidence of electron acceleration at a quasi-parallel shock. Here we use Cassini data to make the first comparison between energy spectra of locally accelerated electrons under these differing upstream magnetic field regimes. We present data taken during a quasi-perpendicular shock crossing on 2008 March 8 and during a quasi-parallel shock crossing on 2007 February 3, highlighting that both were associated with electron acceleration to at least MeV energies. The magnetic signature of the quasi-perpendicular crossing has a relatively sharp upstream–downstream transition, and energetic electrons were detected close to the transition and immediately downstream. The magnetic transition at the quasi-parallel crossing is less clear, energetic electrons were encountered upstream and downstream, and the electron energy spectrum is harder above ~100 keV. We discuss whether the acceleration is consistent with diffusive shock acceleration theory in each case, and suggest that the quasi-parallel spectral break is due to an energy-dependent interaction between the electrons and short, large-amplitude magnetic structures.