Being such a problem for me I've given this some thought in the past.
One possibility is mitochondrial dysfunction. As regards an impaired cold response, the primary physiological response (besides behavioural responses like shivering or a hot whiskey) is via brown adipose (fat) tissue which if I recall correctly is dense in basically 'faulty' mitochondria that, rather than producing energy, light up like an electric fire to produce heat (and also burn off calories as result which is why they were once thought possibly might be the key to tackling obesity). Impaired mito function might show up readily (and possible first) in impaired brown adipose function.
How this explains heat intolerance on the other hand I don't know but heat intolerance is well recognised in pure mitochondrial diseases. Perhaps the physiology of heat dissipation is also a high energy requirement?
Another possibility is autonomic dysfunction/neuropathy which is now increasingly a common finding in ME/CFS. Decoupling (I like that concept) of the autonomic response from cold and heat sensory input might very well explain this inability to maintain thermal homeostasis.
Perhaps the deficit in heat shock proteins found in ME/CFS patients might play a role given that amongst the first HSPs discovered were actually those induced by cold stress?
MS patients also have severe problems with heat intolerance which they report cause symptoms flares. Actually this gels with me because heat just doesn't cause me to feel uncomfortably hot, rather it causes a flare in all of my usual symptoms and can often lead to an attack of episodic ataxia in the same way that physical or mental stressors can. In the case of MS its believed that high temperatures slows the speed of neural signals that is already impaired in MS due to demyelination. As far as I'm aware demyelination has been ruled out in ME/CFS but I do believe we may have other problems with neurotransmission.
One thing that does work for me to an extent (apart from total immersion in cold water which isn't always practical) is the supplement NAC in soluble form. This might tie in with mito dysfunction and possibly even autonomic or neural problems. On the other had the fact that its a sugary drink might just mean I'm replacing electrolytes.
Sorry. Its been a while since I looked into this and to cut a long story short I don't know the answer!
Some clues might be found in other conditions where its a problem though.