The context for our paper comes from the neurophenomenology research program initiated by Francisco Varela at the end of the 1990s. Varela’s working hypothesis was that, to be successful, a consciousness research program must progress by relating first-person phenomenological accounts of the structure of experience and their third-person counterparts in neuroscience through reciprocal or mutual constraints. Leveraging Bayesian mechanics, in particular deep parametric active inference, we demonstrate the potential for epistemically advantageous mutual constraints between phenomenological, computational, behavioural and physiological vocabularies. Specifically, the dual information geometry of Bayesian mechanics serves to establish, under certain conditions, generative passages between lived experience and its physiological instantiation. This paper argues for the epistemological necessity of such passages and the inclusion of trained reflective awareness in neurophenomenological empirical approaches, showcasing incremental epistemic gains that come from by shifting the focus from the contents of experience (i.e. what a subject experiences in a given experimental set-up) to the how of experience - the activities of consciousness that allow for a meaningful world to appear to us as such in lived experience. The explanatory power of the resulting framework, deep computational neurophenomenology, arises from the disciplined circulation between first and third-person perspectives enabled by the formalism of deep parametric active inference, where parametric depth refers to a property of generative models that can form beliefs about the parameters of their own modelling process. Hence, this computational formalism contributes to understanding consciousness by bridging phenomenological descriptions and physiological instantiations, whilst also highlighting the significance of trained first person investigation in experimental protocols.