Contact Hamiltonian mechanics extends the symplectic formalism to dissipative and non-conservative thermodynamic systems through a contact 1-form α on a (2n+1)-dimensional manifold, whose Reeb field R and contact Hamiltonian vector field XH generate the equations of motion. In existing numerical implementations, α and its exterior derivative dα are typically represented only implicitly, through symbolic display strings and hand-coded component formulas, with non-degeneracy checked by a hard-coded factorial rather than an actual computation. We construct α, dα, R, and XH as genuine multivectors in a Grassmann/Clifford algebra Cl(2n + 1, 0, 0), compute the non-degeneracy condition directly as the wedge product α ∧ (dα)n, and show the resulting XH reproduces the standard Bravetti—Cruz—Tapias equations of motion exactly. We report, however, that this base construction uses only the outer product and a signature-independent coordinate pairing: repeating the identical computation across several signature choices of the same dimension, including a degenerate split, leaves every reported quantity unchanged, so the Clifford geometric product itself does no work in this part of the construction — it is exterior (Grassmann) algebra wearing Clifford notation. Motivated by this, we then ask a genuinely open question in a different, previously rejected carrier, the degenerate algebra Cl(n, n, 1) in which dα is itself a hyperbolic metric bivector: can contactomorphisms (the symmetries of the contact structure) be represented as GA versors, exponentials of bivectors applied by the sandwich v 7→ V vV −1? We find a mixed, carefully bounded answer: the symplectic squeeze transformation is realized exactly by a versor sandwich, with the hyperbolic signature doing genuine work (each pair-bivector squares to +1); the Reeb flow and the conformal dilation (for scale factors λ /∈ {±1}), by contrast, are provably not representable as pure versor sandwiches, for structural reasons (an affine translation cannot fix the origin; a radical-rescaling forbidden by a rigidity lemma we prove for the degenerate metric) that we make precise. The complete construction, both the base representation and the versor investigation, together with a fully regression-tested reference implementation, is released as part of the open-source contact-thermodynamics library.