The K-band light profiles from two statistically complete, diameter-limited samples of disc galaxies have been simultaneously modelled with a seeing-convolved Sérsic r1/n bulge and a seeing-convolved exponential disc. This has enabled an accurate separation of the bulge and disc light, and hence an estimate of the central disc surface brightness μ0,K and the disc scalelength h. There exists a bright envelope of galaxy discs in the μ0,K-logh diagram; for the early-type (<=Sbc-Sc) disc galaxies μ0,K is shown to increase with logh, with a slope of ~2 and a correlation coefficient equal to 0.75. This relation exists over a range of disc scalelengths from 0.5 to 10kpc (H0=75kms-1Mpc-1). In general, galaxy types Scd or later are observed to deviate from this relation; they have fainter surface brightnesses for a given scalelength. With a subsample of 59 low-inclination (i<=50°) and 29 high-inclination (i>=50°) galaxies having morphological types ranging from S0 to Sc, the need for an inclination correction to the K-band disc surface brightness is demonstrated. Certain selection criteria biases which have troubled previous surface brightness inclination tests (for example, whether the galaxies are selected from a magnitude- or diameter-limited sample) do not operate in the μ0,K-logh diagram. Measured central disc surface brightnesses are found to be significantly (>5σ) brighter for the high-inclination disc galaxies than for the low-inclination disc galaxies. With no surface brightness inclination correction or allowance for the trend between μ0,K and logh, the standard deviation to the distribution of μ0,K values is ~1magarcsec-2, while the standard deviation about the mean μ0,K-logh relation decreases from 0.69magarcsec-2, when no inclination correction is applied, to 0.47magarcsec-2 when the inclination correction is applied. Possible changes to the disc scalelength with inclination, as a result of radial gradients in the disc opacity, have been explored. The maximum possible sizes for such corrections are too small to provide a valid explanation for the difference between the low- and high-inclination disc galaxies in the μ0,K-logh diagram.