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.

6pp, 4 figures V2: minor typo corrections and text clarifications, matches journal version