  Axially symmetric processes on spheres, for which the second-order dependency
structure may substantially vary with shifts in latitude, are a prominent
alternative to model the spatial uncertainty of natural variables located over
large portions of the Earth. In this paper, we focus on Karhunen-Lo\`eve
expansions of axially symmetric Gaussian processes. First, we investigate a
parametric family of Karhunen-Lo\`eve coefficients that allows for versatile
spatial covariance functions. The isotropy as well as the longitudinal
independence can be obtained as limit cases of our proposal. Second, we
introduce a strategy to render any longitudinally reversible process
irreversible, which means that its covariance function could admit certain
types of asymmetries along longitudes. Then, finitely truncated
Karhunen-Lo\`eve expansions are used to approximate axially symmetric
processes. For such approximations, bounds for the $L^2$-error are provided.
Numerical experiments are conducted to illustrate our findings.
