A double-normal pair of a finite set S of points that spans R-d is a pair of points {p, q} from S such that S lies in the closed strip bounded by the hyperplanes through p and q perpendicular to pq. A double-normal pair {p, q} is strict if S \ {p,q} lies in the open strip. The problem of estimating the maximum number N-d(n) of double-normal pairs in a set of n points in R-d, was initiated by Martini and Soltan [Discrete Math. 290 (2005), 221-228]. It was shown in a companion paper that in the plane, this maximum is 3[n/2], for every n > 2. For d >= 3, it follows from the Erdos-Stone theorem in extremal graph theory that N-d(n) = 1/2(1 - 1/k)n(2) + o(n(2)) for a suitable positive integer k = k(d). Here we prove that k(3) = 2 and, in general, [d/2] <= k(d) <= d - 1. Moreover, asymptotically we have lim(n ->infinity) k(d)/d = 1. The same bounds hold for the maximum number of strict double-normal pairs.