We study the formal and physical aspects of providing particle internal symmetries with a space-time origin. Quantum mechanics impose a requirement of added dimensionality. We develop a formalism introduced by Fronsdal and Joseph, postulating the abstracted existence of a pseudo-Euclidean space in which curved space-time is embedded. We use the global embedding space, without requiring it to be homogeneous or isotropic. Symmetries correspond to available transformations carrying the Riemann space-or part of it-into itself; the Lorentz group does it locally, operating in the Minkowski tangent space. In addition, we now have the Joseph generators corresponding to orthogonal transformations in the normal space. Checking through known local solutions of the Einstein equations, and using recent developments in differential geometry, we find local embeddinge seem to require ten dimensions, with a possibly compact normal subspace. Global embeddings may require more dimensions, though this is not corroborated by the experience gained from embedding the present set of cosmological models. The probable total endosymmetry can be fitted in a ten-dimensional embedder provided we apply analytical continuation in the Joseph space. If we keep to real rotation groups, both plausible chiral schemes require sixteen dimensions in the embedding. The main physical consequence for particle interactions, according to the geometrical model of symmetry-breaking we suggest, would reside in predicting symmetry breakdown in regions of very high curvature, such as may be found in some stellar cores.