Aims. In this study we try to answer the question whether or not subluminous Type Ia supernovae have additional distinctive properties when examined from the point of view of the explosion geometry. Methods. We have performed optical spectropolarimetric observations of the Type Ia SN 2005ke at 3 epochs (days −8, −7, and +76). The explosion properties are derived by comparing the data to explosion and radiation transfer models. Results. The supernova shows polarimetric properties that are very similar to the only other subluminous event for which spectropolarimetry is available, i.e. SN 1999by. The data present a very marked dominant axis, which is shared by both the continuum and lines such as Si ii λ6355, suggesting that the relatively large, global asymmetry is common to the photosphere and the line-forming region. The maximum polarization degree observed in the Si ii λ6355 absorption reaches 0.39 ± 0.08%. At variance with what is seen in core-normal Type Ia, SN 2005ke displays significant continuum polarization, which grows from the blue to the red and peaks at about 7000 Å, reaching ∼0.7%. The properties of the polarization and flux spectra can be understood within the framework of a subluminous delayed-detonation (DD), or pulsating DD scenario, or white dwarf (WD) mergers. The difference in appearance with respect to core-normal SNe Ia is caused by low photospheric temperatures in combination with layers of unburned C, and more massive layers of the products of explosive C and O burning. The comparatively high level of continuum polarization is explained in terms of a significant global asymmetry (∼15%), which is well reproduced by an oblate ellipsoidal geometry within the general context of a DD explosion. Conclusions. Our results suggest that SN 2005ke arose either from a single-degenerate system in which the WD is especially rapidly rotating, close to the break-up velocity, or from a double-degenerate merger. Based on the current polarization data, we cannot distinguish between these two possibilities. Possible tests are discussed.