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We report a new measurement of transverse single-spin asymmetries for dijet production in collisions of polarized protons at $\sqrt{s}=200\mathrm{GeV}$. Correlations between the proton spin and the transverse momenta of its partons, each perpendicular to the proton momentum direction, are probed at high $Q^2\approx 160{\mathrm{GeV}}^2$. Evidence for nonzero Sivers effects is measured for the first time in dijets from proton-proton collisions, but only when the jets are sorted by their net charge, which enhances the otherwise canceling opposite-sign $u$- or $d$-quark contributions to separate data samples. The resulting asymmetries are compared to recent theoretical calculations. Separately, the associated Sivers observable $⟨k_T⟩$, the average parton transverse momentum, is extracted using a simple kinematics approach which further enables a determination of the individual partonic contributions to the observed asymmetries.