We report the first measurements of transverse single-spin asymmetries for inclusive jet and jet + $pi^{pm}$ production at midrapidity from transversely polarized proton-proton collisions at $sqrt{s} = 500$ GeV. The data were collected in 2011 with the STAR detector sampled from 23 pb$^{-1}$ integrated luminosity with an average beam polarization of 53%. Asymmetries are reported for jets with transverse momenta $6 < p_{T, jet} < 55$ GeV/c and pseudorapidity $|eta| < 1$. Presented are measurements of the inclusive-jet azimuthal transverse single-spin asymmetry, sensitive to twist-3 initial-state quark-gluon correlators; the Collins asymmetry, sensitive to quark transversity coupled to the polarized Collins fragmentation function; and the first measurement of the Collins-like asymmetry, sensitive to linearly polarized gluons. Within the present statistical precision, inclusive-jet and Collins-like asymmetries are small, with the latter allowing the first experimental constraints on gluon linear polarization in a polarized proton. At higher values of jet transverse momenta, we observe the first non-zero Collins asymmetries in polarized-proton collisions, with a statistical significance of greater than $5sigma$. The results span a range of x similar to results from SIDIS but at much higher $Q^{2}$. The Collins results enable tests of universality and factorization-breaking in the transverse momentum-dependent formulation of perturbative quantum chromodynamics.
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