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We present ALMA observations of CO $J = 2-1$ and CS $J = 5-4$ emission from the disk around TW~Hydrae. Both molecules trace a predominantly Keplerian velocity structure, although a slowing of the rotation velocity is detected at the outer edge of the disk beyond ${\approx}~140$~au in CO emission. This was attributed to the enhanced pressure support from the gas density taper near the outer edge of the disk. Subtraction of an azimuthally symmetric background velocity structure reveals localized deviations in the gas kinematics traced by each of the molecules. Both CO and CS exhibit a `Doppler flip' feature, centered nearly along the minor axis of the disk (${\rm PA} \sim 60\degr$) at a radius of $1\farcs35$, coinciding with the large gap observed in scattered light and mm~continuum. In addition, the CO emission, both through changes in intensity and its kinematics, traces a tightly wound spiral, previously seen with higher frequency CO $J = 3-2$ observations (Teague et al., 2019). Through comparison with linear models of the spiral wakes generated by embedded planets, we interpret these features in the context of interactions with a Saturn-mass planet within the gap at a position angle of ${\rm PA} = 60\degr$, consistent with the theoretical predictions of (Mentiplay et al. 2019). The lack of a corresponding spiral in the CS emission is attributed to the strong vertical dependence on the buoyancy spirals which are believed to only grow in the atmospheric of the disk, rather than those traced by CS emission.