Daisy Chaining different shift registers.

Hi guys, I was hoping one of you could tell me if it would be possible to daisy chain a parallel-in/serial-out shift register with a serial-in/parallel-out shift register. I would essentially want to drive 8 LED's with the serial-in/parallel-out IC (595), and read in 8 buttons with the parallel-in/serial-out IC (4021) using only 3 pins since I plan on using an ATTiny 45 for this and so my pins are quite limited.
Thanks!

off the top of my head I think that could become a mess, since they would be on the same lines when you read it back it might get read in by the S-P register and mess up your LED display. (I guess if you are updating fast enough you may not notice it)

Honestly that is why they have a chip select pin, and you might be able to do some transistor setup to toggle between the two while only using 1(more) avr pin