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 (13\phantom{\rule{0.25em}{0ex}}\text{TeV}), corresponding to an integrated luminosity of (137\phantom{\rule{0.25em}{0ex}}{\text{fb}}^{-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.