  Measurements of the $W^\pm$ mass ($m_W$) provide an important consistency
check of the Standard Model (SM) and constrain the possibility of physics
beyond the SM. Precision measurements of $m_W$ at hadron colliders are inferred
from kinematic distributions of transverse variables. We examine how this
inference is modified when considering the presence of physics beyond the SM
expressed in terms of local contact operators. We show that Tevatron
measurements of $m_W$ using transverse variables are transparent and applicable
as consistent constraints in the Standard Model Effective Field Theory (SMEFT)
with small measurement bias. This means that the leading challenge to
interpreting these measurements in the SMEFT is the pure theoretical
uncertainty in how these measurements are mapped to Lagrangian parameters. We
stress the need to avoid using naive combinations of Tevatron and LEPII
measurements of $m_W$ without the introduction of any SMEFT theoretical error
to avoid implicit UV assumptions. In a companion paper, we implement our
procedure to consistently incorporate $m_W$ measurements into a global fit.
