Line-of-sight velocities of gas and stars can constrain dark matter (DM) within rotationally supported galaxies if they trace circular orbits extensively. Photometric asymmetries may signify non-circular motions, requiring spectra with dense spatial coverage. Our integral-field spectroscopy of 178 galaxies spanned the mass range of the Sydney-AAOMulti-object integral field spectrograph (SAMI) Galaxy Survey. We derived circular speed curves (CSCs) of gas and stars from non-parametric fits out to r ~ 2re. For 12/14 with measured HI profiles, ionized gas and HI maximum velocities agreed. We fitted mass-follows-light models to 163 galaxies by approximating the radial light profile as nested, very flattened mass homeoids viewed as a Sérsic form. Fitting broad-band spectral energy distributions to Sloan Digital Sky Survey images gave median stellar mass/light 1.7 assuming a Kroupa initial mass function (IMF) versus 2.6 dynamically. Two-thirds of the dynamical mass/light measures were consistent with star+remnant IMFs. One-fifth required upscaled starlight to fit, hence comparable mass of unobserved baryons and/or DM distributed like starlight across the SAMI aperture that came to dominate motions as the starlight CSCs declined rapidly. The rest had mass distributed differently from light. Subtracting fits of Sérsic radial profiles to 13 VIKING Z-band images revealed residual weak bars. Near the bar major axis, we assessed m = 2 streaming velocities, and found deviations usually <30 km s-1 from the CSC; three showed no deviation. Thus, asymmetries rarely influenced the CSC despite colocated shock-indicating, emission-line flux ratios in more than 2/3 of our sample.