(where “this” = unusual heat/temperature relationship)
When two objects (one hotter than the other) come into thermal contact, energy will flow between them until the global entropy is maximized. Note that the directionality of flow cares about entropy and not energy content. Thermal equilibrium is reached when the change in entropy per unit change of internal energy is equal between the two subsystems. (Otherwise, the sum of the entropies could be made higher still by moving a bit of energy from one subsystem to the other.)
In “normal” systems, increasing the internal energy increases the entropy. In a discrete system (often realized with spin states) or a cleverly constrained system where there is a maximum internal energy, you can have the number of available internal states start to decrease as internal energy increases, and thus so too does the entropy – which is the logarithm of the number of available internal states – decrease as the internal energy increases. In those systems, energy needs to flow out to increase the entropy. And that’s what matters.
The italicized text two paragraphs above is the fundamental definition of the inverse of temperature. So, 1/(temperature) is defined as “change in entropy per unit change of internal energy”, and so the latter being equal between two systems in thermal equilibrium is the same as saying the temperature is equal. But, since entropy might decrease with energy added, temperature can be negative.
But it also means that temperature (T) is mathematically sort of the wrong quantity to use when talking about equilibrium and energy flow. 1/T or 1/(k_B T), usually written \beta, handles this more cleanly.
Let “cold” and “hot” relate to how energy will flow (i.e., from hot to cold). Then T=+0 is the coldest you can get, and then T=\infty is a lot hotter, and then T=-\infty is hotter still, and then T=-0 is the hottest you can get (where suitable limits should obviously be taken in this notation).
So, any negative temperature is hotter than infinitely positive temperature. And, say, -1 Kelvin is way hotter than 1 Kelvin. Which is to say: a system as -1 K will very happily give up heat to a system at 1 K.
If we were to use \beta instead of T, the language gets a lot cleaner. Higher \beta means colder; lower \beta means hotter. Full stop. One is free to talk about -\beta instead to flip the axis over if you want hotter to run in the positive direction, but that’s not the usual convention for \beta, and there’s no particular need to do so.
In summary, \beta is just “change in entropy for a change in energy”, and that can be positive or negative for a given system and that “slope” varies as energy changes, and energy flow ceases when \beta is equal between two systems in thermal contact, and energy always flows from the lower \beta system to the higher \beta system.
When negative \beta's are involved, it can no longer be said that “energy always flows from higher T to lower T”, but that’s mostly because T was a poor choice in the first place.
But, for everyday systems, energy is unbounded and/or there are suitable “continuous” degrees of freedom such that entropy is always strictly increasing with energy, and so we typically experience only the positive temperature region.