I've come to understand that twisted curves, as for instance defined in the Brainpool specifications, are $F(p)$-isomorphic to their regular $F(p)$ equivalents. So brainpoolP256r1 is isomorphic to brainpoolP256t1.
In a protocol, does it make sense to specify support for twisted curves? Should it be possible to specify the twisted curve OID instead of the $F(p)$ equivalent? If so, could using the twisted curve representation of the curve have any performance benefits over the regular representation? Or is the twisted curve representation only used for the security analysis of the curve?