  We study a quasimorphism, which we call the Dehn twist coefficient (DTC),
from the mapping class group of a surface (with a chosen compact boundary
component) that generalizes the well-studied fractional Dehn twist coefficient
(FDTC) to surfaces of infinite type. Indeed, for surfaces of finite type the
DTC coincides with the FDTC. We provide a characterization of the DTC (and thus
also of the FDTC) as the unique homogeneous quasimorphism satisfying certain
positivity conditions.
  The FDTC has image contained in $\mathbb{Q}$. In contrast to this, we find
that for some surfaces of infinite type the DTC has image all of $\mathbb{R}$.
To see this we provide a new construction of maps with irrational rotation
behavior for some surfaces of infinite type with a countable space of ends or
even just one end. In fact, we find that the DTC is the right tool to detect
irrational rotation behavior, even for surfaces without boundary.
