[jbay808]

Yes, this is in accordance with the definition of temperature as d entropy / d energy, which is the more generally-applicable definition. It's easy to see from this that temperature becomes negative if adding more energy into the system causes the entropy to decrease. In most systems that doesn't happen because there are always an increasing number of higher-energy states that open up, but in a very carefully constrained system, the added energy forces the system into a smaller number of excited states away from a higher-entropy ground state. A real-world example of this is a laser's population inversion.

A system at finite negative temperature is actually considered "hotter" than a normal system at any positive finite temperature; if you put them in thermal contact, heat will flow in the direction that increases entropy, which you get by taking energy out of the negative-temperature inverted system and adding it into the ordinary positive-temperature system. This increases the entropy of both systems.

The definition where temperature is the "average kinetic energy of the particles" is a special case, and it only really works when that energy is evenly distributed over all degrees of freedom. For example, you wouldn't consider an icy comet to be at a high temperature just because it's moving quickly, even though its particles have a great deal of kinetic energy on average!