  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 $5\sigma$. 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.
