We develop a systematic study of Jahn-Teller (JT) models with continuous symmetries by explor- ing their algebraic properties. The compact symmetric spaces corresponding to JT models carrying a Lie group symmetry are identified, and their invariants used to reduce their adiabatic potential energy surfaces into orbit spaces. Each orbit consists of a set of JT distorted molecular structures with equal adiabatic electronic spectrum. Molecular motion may be decomposed into pseudorota- tional and

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... ial. The former preserves the orbit, while the latter maps an orbit into another. The dimensionality and topology of the internal space of each orbit depends on the number of degener- ate states in its adiabatic electronic spectra. Furthermore, qualitatively different pseudorotational modes occur in orbits of different types. We also provide a simple proof that the electronic spectrum for the space of JT minimum-energy structures (trough) displays a universality predicted by the epikernel principle. This result is in turn used to prove the topological equivalence between bosonic (fermionic) JT troughs and real (quaternionic) projective spaces, a conclusion which has outstanding physical consequences, as explained in our work. The relevance of our study for the more common case of JT systems with only discrete point group symmetry, and for generic asymmetric molecular systems with conical intersections involving more than two states is likewise discussed. In particular, we show that JT models with continuous symmetries present the simplest models of conical intersections among an arbitrary number of electronic state crossings.