Isospin symmetry is a fundamental concept in nuclear physics. Even though isospin symmetry is partially broken, it holds approximately for most nuclear systems, which makes exceptions very interesting from the nuclear structure perspective. In this framework, it is expected that the spins and parities of the ground states of mirror nuclei should be the same, in particular for the simplest systems where a proton is exchanged with a neutron or vice versa. In this Letter, we present evidence that this assumption is broken in the mirror pair 71Br and 71Kr system. Our conclusions are based on a high-statistics /3 decay study of 71Kr and on state-of-the-art shell model calculations. In our work, we also found evidence of a new state in 70Se, populated in the /3-delayed proton emission process which can be interpreted as the long sought coexisting 0 & thorn; state.