How many latent variables are needed to fit the cosmic microwave background (CMB) temperature and polarisation power spectra when a neural network is given no cosmological model, and to what extent do they map onto the standard cosmological parameters? We use multi-task learning with variational autoencoders trained on CLASS-generated spectra (TT, TE, EE) over 2≤ℓ≤2500. No cosmological model is built into the network, and the number of latent variables, L, is varied over L=5, 6 and 7. Their fits are compared to the Planck 2018 data using χ̂², AIC, BIC and Bayesian evidence. Their correspondence with cosmological parameters is measured with normalised mutual information post-training. The loss weights are set within a single training run by uncertainty weighting, with the weight of the disentanglement term either fixed (semi-automated, SA) or learned (completely automated, CA). The reconstruction loss is split over multipole ranges in three configurations that separate the roles of temperature and polarisation and of the low- and high-ℓ ranges. In four of the six combinations of configuration and strategy, the fit improves from L=5 to L=6 and not beyond, and L=6 is the best, or close to the best, in five of the six cases. So, six latent variables are needed, as many as the free parameters of the standard, model-dependent approach. SA gives clearer disentanglement, while CA needs no a priori choice of the disentanglement weight and has the higher evidence at L=5 and 6: the two strategies are points on a trade-off between interpretability and fit to the data. Notably, the latent variables for τ and Aₛ share almost no information in any model, consistent with large-scale polarisation separating the two, as in standard analyses. A deep-learning approach can thus give a compact, data-driven parametrisation of the CMB spectra.