Mystery solved: My homemade development board has 3 other PIC sockets, 2 of which are occupied with 2 different PICs, running different programs. All of these parts share the same switch. This switch has a 5K resistor to ground, and pulls to +5V. The 16F688 has it's external interrupt pin connected to the switch, and also had it's internal pullups enabled. With only the 12F629 attached (internal pullups enabled), the voltage on the INT pins was +1V. With both PICS attached, the voltage rises to +2V. The external interrupt pins are Schmitt trigger inputs. The data sheet says the minimum voltage for "a 1, or a high" is 0.8V.
So the 12F629 was seeing a high when it should have been a low. I have the software set to trigger on the rising edge of the switch, so it went HIGHER, LOWER, and then HIGHER. With both PICS having their internal pullups disabled, the voltage is now +2mv, and all is well. So much for "weak" pullups! As usual, I learned a lesson. Thanks to everyone for your help.