Can I cut off a thermocouple's metal head?
Yes, you should be able to cut the wires and reattach them without losing calibration.  I wouldn't necessarily just twist them up, though.  A weld would be best, and brazing second best (http://eleceng.dit.ie/gavin/Instrument/Temperature/Laws%20of%20ThermoC.html -- the law of intermediate metals can be used to show this).  A blob of solder is sort of like a braze.  Even a bad solder is likely good enough.

Twisting will give you some life, but it may not be gas-tight.  I'd be concerned about oxidation eventually causing intermittent connections.  Crimping should work too, in a pinch, and there are gas-tight ways to crimp. 

https://www.physicsforums.com/threads/okay-to-twist-thermocouple-wires.793193/#post-4982730

Reference https://www.physicsforums.com/threads/okay-to-twist-thermocouple-wires.793193/