Here we report on the precise structural investigation of multiwalled boron nitride (BN) nanotubes by means of high-resolution transmission electron microscopy and electron energy loss spectroscopy. The nanotubes were produced from carbon nanotubes by applying a recently discovered technique: a substitution chemical reaction [W. Han, Y. Bando, K. Kurashima, and T. Sato, Appl. Phys. Lett. 73, 3085 (1998)]. It is found that in contrast to the starting carbon nanotubes, which exhibited large number of shells (typically >10), a significant proportion of buckled and corrugated graphene-like sheets, poor degree of graphitization, and wide distribution of helicities, the resultant BN nanotubes revealed perfectly straight shapes, limited number of shells (typically 2-6), and remarkable ordering of the graphene-like sheets in the so-called nonhelical "zig-zag" fashion with the [101̄0] direction parallel to the tube axis.