We report the first measurement of the opening angle distribution between pairs of jets produced in high-energy collisions of transversely polarized protons. The measurement probes (Sivers) correlations between the transverse spin orientation of a proton and the transverse momentum directions of its partons. With both beams polarized, the wide pseudorapidity (-1< or = eta < or = +2) coverage for jets permits separation of Sivers functions for the valence and sea regions. The resulting asymmetries are all consistent with zero and considerably smaller than Sivers effects observed in semi-inclusive deep inelastic scattering. We discuss theoretical attempts to reconcile the new results with the sizable transverse spin effects seen in semi-inclusive deep inelastic scattering and forward hadron production in pp collisions.