research level - Why aren't the spin-3/2 fields in the (3/2,0)+(0,3/2) representation?

Why is it that spin-$\frac 32$ fields are usually described to be in the $(\frac 12, \frac 12)\otimes[(\frac 12,0)\oplus(0,\frac 12)]$ representation (Rarita-Schwinger) rather than the $(\frac 32,0)\oplus(0,\frac 32)$ representation? Does the latter not describe a spin-$\frac 32$ field? Why is the gravitino given by the Rarita-Schwinger-type representation rather than the $(\frac 32,0)\oplus(0,\frac 32)$ representation?

This is related to a recent question I asked on gauge invariance of the Rarita-Schwinger field.

Thanks!