The 32x was developed by Sega of America at the same time the Saturn was in development at Sega of Japan. Apparently Sega had communication issues. I own a Sega CD, Saturn and Dreamcast. The 32x was really a strange beast as it required a Sega Genesis. It plugged into the top of the Genesis and allowed you to play 32 bit cartridges or if you had a Sega CD, you could play 32x CDs.
The Genesis (which just turned 20 this month by the way) was an odd system in that with a Power Base converter you could play 8-bit Master System games too. So you could downgrade the Genesis to a 3rd gen 8-bit system, play 4th gen Genesis/SegaCD games or 5th gen 32x games. 8-bit master system games could be run on the Genesis audio hardware. The Master System contained a Z80 cpu with a TI "Programmable Sound Generator" and a custom video chip. The Genesis had a 68000 CPU, which handed the Z80 all of the audio duties. The Z80 still had control of the TI PSG as well as a Yamaha YM2612 4op FM synth chip. The video chip in the Genesis was essentially the same as the Master System except it had new screen modes for Genesis mode. Master System emulation was a breeze for the Genesis.
Wow sorry I got to rambling! I am just fascinated by Sega hardware. Interestingly, Sega's home consoles were all based on their arcade boards. The Genesis board was based on the 16-bit Musashi board, and the DC was based on Naomi.
Happy Birthday Dreamcast!