hi,
My question is how will the device know whether it is being read or written, when the Din and Dout pins are tied together?

I see, you have read the manual; try do it once again, for example, page 25:

All communications to the part must start with a write operation to the Communications Register. The data written to the Communications Register determines whether the next operation is a read or write operation, the type of read operation, and on which register this operation takes place.

So you question has well documented answer inside datasheet, just read it carefully.

Regards,
Oleg