While following Moen's instructions and twisting the old cartridge to remove it, its shell broke leaving a segment jammed deep inside. The cartridge had been in place for 26 years. With considerable difficulty, I extracted the two rubber sheet-type pieces. How do I remove the hard plastic portion of the jammed segment?

Run a long wood screw into the remnants of the cartridge, then pull the screw and what's left of the cartridge out.