hi,
oops, I have missed that your chip has two serial ports.
Well, so the answer depends on protocol that DSP uses with. If it looks like two shift registers (one for input and another for output) and DSP is passive device which are clocked externaly (!!!) then it is possible, indeed. But if any handshake/start conditions presents or if DSP clocks data out itself then it is not possible with mode 0. There is not enough information you provided to answer clean.

Regards,
Oleg