Is there a family of functions that includes triangle, sin, and square waves? Is there a family of functions that includes triangle, sin, and square waves?
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If so, is there a way to parametrise them such that a single parameter sweeps from triangle through sin to square? Something like how kurtosis sweeps through the t-distributions?

Finally, if the answer is 'yes' to both of the above, then is there something equivalent to skewness that would result in a saw wave, when the first parameter was set to a triangle wave?

Using linear combination of $\sin$ functions and $\cos$ you can express any periodic function
$$f(x)=\sum_{n=0}^{+\infty}a_n\cos nx+b_n\sin nx$$
It's called Fourier series
http://mathworld.wolfram.com/FourierSeries.html
I'm not sure about single parameter sweep. You surely can transform between them freely with linear transformations (all these functions simply have different coordinates in the basis of $\cos$ and $\sin$ functions)
