What is $5^\circ \mathrm{F}$ minus $5^\circ \mathrm{F}$? Is it $0^\circ \mathrm{F}$ or is it $0\,\mathrm{K}$ (Kelvin)?
From an arithmetic standpoint, it seems like it should be $0^\circ \mathrm{F}$, but that seems inconsistent because the result represents a delta ($0 \,\Delta^\circ \mathrm{F}$ perhaps) which is conceptually completely different. Doing math with units like $^\circ\mathrm{F}/s$ completely breaks down.
From a natural sciences standpoint, it's 0 K, because an amount of heat minus the same amount of heat is no heat, but that seems inconsistent because everyone knows that $5 - 5 = 0$, and this would instead yield $5^\circ \mathrm{F} - 5^\circ\mathrm{F} = -459.67^\circ\mathrm{F}$, which may be surprising.
Is there a convention for doing arithmetic on Interval Scale measurements? That same page suggests there is a difference operation for them in its comparison table.
For context, I'm working on a Python library for doing dimensioned arithmetic (with an eye towards the natural sciences) and interval scales are making things complicated. For instance, $^\circ\mathrm{F}\cdot s$ doesn't seem to represent a meaningful physical dimension; it can't be converted to $\mathrm{K}\cdot s$. For example, trying to convert $6^\circ\mathrm{C}\cdot s$ by visualizing temperature and time as axes:


There have been great answers so far, but I'd like to focus on the practical implementation side of this. Which operations do you implement and how?
Temperature forms an affine line, no matter which unit you're using. It is much like with dates and time: you cannot add last Tuesday to 17 October of 2015, what would that even mean? And so Python does not let you add two instances of datetime. You can subtract them, though, giving you a timedelta object.
Similarly, you would need to implement a delta-temperature unit to represent differences in temperature. This delta-temperature can then implement the same operations as timedelta: you can add them together or subtract them, you can scale them by a scalar, you can divide two delta-temperatures to get a scalar value, and you can add or subtract from a temperature value.
