You just use the 8bit stream and read/write 4 bytes at a time or use 16bit and read/write 2*16bits=4 bytes. You don't need "32bit SPI". It will work fine.
Thus, the processor accepts 16 bits, on interruption I pull them out from the buffer, and further there is a reception of following 16 bits, again interruption. Correctly?
Adjustment DMA in such mode is possible?