We present seven epochs of spectropolarimetry of the Type IIb supernova (SN IIb) 2011dh in M51, spanning 86 d of its evolution. The first epoch was obtained 9 d after the explosion, when the photosphere was still in the depleted hydrogen layer of the stripped-envelope progenitor. Continuum polarization is securely detected at the level of P ≈ 0.5 per cent through day 14 and appears to diminish by day 30, which is different from the prevailing trends suggested by studies of other core-collapse SNe. Time-variable modulations in P and position angle are detected across P-Cygni line features. H α and He i polarization peak after 30 d and exhibit position angles roughly aligned with the earlier continuum, while O i and Ca ii appear to be geometrically distinct. We discuss several possibilities to explain the evolution of the continuum and line polarization, including the potential effects of a tidally deformed progenitor star, aspherical radioactive heating by fast-rising plumes of 56Ni from the core, oblique shock breakout, or scattering by circumstellar material. While these possibilities are plausible and guided by theoretical expectations, they are not unique solutions to the data. The construction of more detailed hydrodynamic and radiative-transfer models that incorporate complex aspherical geometries will be required to further elucidate the nature of the polarized radiation from SN 2011dh and other SNe IIb.