Supersymmetric theories with the same bosonic content but different fermions, aka twins, were thought to exist only for supergravity. Here we show that pairs of super conformal field theories, for example exotic N = 3 and N = 1 theories in D = 4 space-time dimensions, can also be twin. We provide evidence from three different perspectives: (i) a twin S-fold construction, (ii) a double-copy argument and (iii) by identifying candidate twin holographically dual gauged supergravity theories. Furthermore, twin W-supergravity theories then follow by applying the double-copy prescription to exotic super conformal field theories.