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This is the second paper of two reporting results from a study of the HI content and stellar properties of nearby galaxies detected by the Arecibo Legacy Fast ALFA blind 21-cm line survey and the Sloan Digital Sky Survey in a 2160 deg^2 region covered by both surveys. We apply strategies of multivariate data analysis to a complete HI flux-limited subset of 1624 objects extracted from the control sample of HI emitters assembled by Toribio et al. (2011a) in order to: i) investigate the correlation structure of the space defined by an extensive set of observables describing gas-rich systems; ii) identify the intrinsic parameters that best define their HI content; and iii) explore the scaling relations arising from the joint distributions of the quantities most strongly correlated with the HI mass. The principal component analysis performed over a set of five galaxy properties reveals that they are strongly interrelated, supporting previous claims that nearby HI emitters show a high degree of correlation. The best predictors for the expected value of MHI are the diameter of the stellar disk, D25r, followed by the total luminosity (both in the r-band), and the maximum rotation speed, while morphological proxies such as color show only a moderately strong correlation with the gaseous content attenuated by observational error. The simplest and most accurate prescription is log(MHI/Msun)= 8.72 + 1.25*log(D25r/kpc). We find a slope of $-8.2 \pm 0.5$ for the relation between optical magnitude and log rotation speed, in good agreement with Tully-Fisher studies, and a log slope of $1.55 \pm 0.06$ for the HI mass-optical galaxy size relation. Given the homogeneity of the measurements and the completeness of our dataset, the latter outcome suggests that the constancy of the average (hybrid) HI surface density advocated by some authors for the spiral population is a crude approximation.