Richard Schwartz, in Mostly Surfaces (Vol. 60. American Mathematical Soc., 2011), defines (on p.14) a translation surface as "a Euclidean cone surface, all of whose 'angle errors' are integer multiples of $\pi$." And a Euclidean cone surface is "a surface that is flat except at finitely many cone points." The 'angle error' is the angle deficit, $2 \pi - \theta(p)$ at the cone point $p$.
My question is:
Q. What role does "integer multiples of $\pi$" play in the theory surrounding translation surfaces? Would half-integer multiples of $\pi$ be less interesting? Rational multiples of $\pi$?
I ask this without understanding much of the theory of translation surfaces. Thanks for any insights the more knowledgeable can provide!