Broken 32kHz Oscillator
If the metal can of oscillator is not grounded, then it works as a capacitive load parallel to crystal. The MHz bursts are probably matching the time delay of second positive feedback of invertor when this capacitance is involved.

http://www.telluriantech.com/Effects_of_Grounding_on_Ceramic_SMD_Crystal.pdf

Replace crystal with 2-10pF variable cap and see if the oscillations will match the frequency of bursts. 

Answer: Ground the can. It will move bursts to GHz. Also the crystal pins can be grounded through small capacitors.