Send and receive bit in one clock

Why can't you use another pin? The SPI interface is specifically designed to use a minimal amount of pins, i.e. 4. If it could have easily been done with 3, I'm sure it would have. I think the straightforward answer is that it's not possible, as you can't have two drivers on a single line. Now, you could probably time multiplex them, so that you know one signal drives first, is captured, and the other drives so much later and is captured so much later that they don't interfere, but you're basically trying to sneak two signals into one clock. You could use rising and falling clock edges for this.