We present infrared spectrophotometry of the Pa alpha (n = 4-3) emission line of hydrogen, together with optical spectrophotometry of H alpha and H beta, of a flux-limited sample of 11 radio sources. The sample consists of all FR II 3CR radio sources with 0.1 less than or equal to z < 0.2 and 5(h) < R.A. < 16(h) which contains eight narrow-line radio galaxies (3CR 135, 184.1, 219, 223, 236, 319, 321, and 327), two broad-line radio galaxies (3CR 234 and 3CR 303), and one quasar (3CR 273). The observations were aimed at the detection of obscured broad lines, hidden from our view by dust absorption, as a means of testing theories of the unification of radio galaxies and quasars. All the objects except 3CR 236 and 3CR 273 show significant reddening of the narrow and broad lines, typically of order A(V) similar to 1.5 for narrow lines and similar to 3 for broad lines. We detect highly obscured broad-line regions in 3CR 184.1, 219, and 223, which appear to be narrow-line objects in the optical, so these should be reclassified as broad-line radio galaxies. In all cases except 3CR 273 and 3CR 303, the broad lines are reddened more than the narrow lines, locating much of the dust responsible for absorbing the broad-line emission between the broad- and narrow-line regions. The dereddened line luminosities range up to those of low-luminosity quasars. The results are broadly consistent with models which seek to unify radio galaxies and quasars through orientation, where an axisymmetric equatorial obscuring region hides the quasar nucleus from view unless the radio axis is pointing close to our line of sight. These data provide the first opportunity to model the distribution of broad-line region extinctions in a complete sample, rather than model just the fraction of quasars and radio galaxies. We develop a simple unification model that matches the observed distribution of extinctions, explains our observations, and makes predictions about the fraction of obscured quasars that will be present in samples of higher radio luminosity.