An important and perhaps critical clue to the mechanism driving the explosion of massive stars as supernovae is provided by the accumulating evidence for asymmetry in the explosion. Indirect evidence comes from high pulsar velocities, associations of supernovae with long-soft gamma-ray bursts, and asymmetries in late-time emission-line profiles. Spectropolarimetry provides a direct probe of young supernova geometry, with higher polarization generally indicating a greater departure from spherical symmetry. Large polarizations have been measured for 'stripped-envelope' (that is, type Ic; ref. 7) supernovae, which confirms their non-spherical morphology; but the explosions of massive stars with intact hydrogen envelopes (type II-P supernovae) have shown only weak polarizations at the early times observed. Here we report multi-epoch spectropolarimetry of a classic type II-P supernova that reveals the abrupt appearance of significant polarization when the inner core is first exposed in the thinning ejecta (approximately 90 days after explosion). We infer a departure from spherical symmetry of at least 30 per cent for the inner ejecta. Combined with earlier results, this suggests that a strongly non-spherical explosion may be a generic feature of core-collapse supernovae of all types, where the asphericity in type II-P supernovae is cloaked at early times by the massive, opaque, hydrogen envelope.