We report the first measurements of the transverse (Px and Py) and longitudinal (Pz) components of the polarization transfer to a bound proton in the deuteron via the H2(e→,e′p→) reaction, over a wide range of missing momentum. A precise determination of the electron beam polarization reduces the systematic uncertainties on the individual components to a level that enables a detailed comparison to a state-of-the-art calculation of the deuteron using free-proton electromagnetic form factors. We observe very good agreement between the measured and the calculated Px/Pz ratios, but deviations of the individual components. Our results cannot be explained by medium modified electromagnetic form factors. They point to an incomplete description of the nuclear reaction mechanism in the calculation.