The first measurements of production cross sections of polarized same-sign W^±W^± boson pairs in proton-proton collisions are reported. The measurements are based on a data sample collected with the CMS detector at the LHC at a center-of-mass energy of 13TeV, corresponding to an integrated luminosity of 137fb^-1. Events are selected by requiring exactly two same-sign leptons, electrons or muons, moderate missing transverse momentum, and two jets with a large rapidity separation and a large dijet mass to enhance the contribution of same-sign W^±W^± scattering events. An observed (expected) 95% confidence level upper limit of 1.17 (0.88) fbis set on the production cross section for longitudinally polarized same-sign W^±W^± boson pairs. The electroweak production of same-sign W^±W^± boson pairs with at least one of the W bosons longitudinally polarized is measured with an observed (expected) significance of 2.3 (3.1) standard deviations.