Infectious disease outbreaks have precipitated a profusion of mathematical models. We introduce a unifying concept of "epidemic momentum" -- prevalence weighted by the capacity to infect in the future -- and use it to reveal a common underlying geometry that corresponds to contours of a generic first integral. Exploiting this conserved quantity, we show that it is possible to (i) disentangle the basic reproduction number $R_0$ from the population proportion that was immune before a disease invasion or re-emergence and (ii) infer both from observed data. This separation enables us to revise the classical estimate of the epidemic final size, incorporating prior population immunity. To illustrate the utility of these insights, we present a novel reappraisal of the main wave of the 1918 influenza pandemic.