Gastruloids are embryonic cell aggregates that elongate themselves from spheres into rods. They thus model formation of the primary body axis of vertebrate embryos. This requires breaking symmetry to organise cell rearrangements known as convergent extension (CE). Gastruloid elongation demonstrates that CE can be entirely self-organising, and that self-organising mechanisms are likely embedded in this and the many other instances of CE in development. Classically, self-organisation in biology involves Turing Reaction-Diffusion (RD) patterning by diffusible morphogens. We have explored the possibility that a completely different, non-RD-based, self-organisation principle for symmetry-breaking by CE could exist, namely polarity-propagating mechanical feedback, which provides nematic structure and for which there is experimental evidence at the single-cell level. Using in silico modelling, we show that this simple rule for mechanical interactions is sufficient to organise convergent extension in 2D whereas, in 3D, we find that a combination of nematic and dipolar organisation, representing Planar Cell Polarity pathway function, gives robust elongation. Thus, although Turing himself chose to consider only chemical morphogenesis, short-range mechanical self-organisation seems likely to be involved in CE in vivo.