[math-fun] Partitioning 3-space into straight lines, exactly one in each direction

Recently someone asked on sci.math.research whether R^3 could be foliated by straight lines, with exactly one line in each direction. Someone else soon provided a proof that this is impossible (using homotopy theory). If we drop the continuity requirement for a foliation, the problem becomes this one: QUESTION: Is R^3 the disjoint union of [bi-infinite] straight lines L_d such that each direction d occurs exactly once? (I.e., each line through the origin of R^3 is parallel or equal to exactly one of the L_d's.) My guess is yes, but it's just a guess. --Dan