Dissipative solitons are self-localized structures that can persist indefinitely in open systems driven out of equilibrium. They play a key role in photonics, underpinning technologies from mode-locked lasers to microresonator optical frequency combs. Here we report on experimental observations of spontaneous symmetry breaking of dissipative optical solitons. Our experiments are performed in a nonlinear optical ring resonator, where dissipative solitons arise in the form of persisting pulses of light known as Kerr cavity solitons. We engineer symmetry between two orthogonal polarization modes of the resonator and show that the solitons of the system can spontaneously break this symmetry, giving rise to two distinct but co-existing vectorial solitons with mirror-like, asymmetric polarization states. We also show that judiciously applied perturbations allow for deterministic switching between the two symmetry-broken dissipative soliton states. Our work delivers fundamental insights at the intersection of multi-mode nonlinear optical resonators, dissipative structures, and spontaneous symmetry breaking, and expands upon our understanding of dissipative solitons in coherently driven Kerr resonators.