  Based on geometric considerations, longitudinal and transverse Lagrangian
velocity increments are introduced as components along, and perpendicular to,
the displacement of fluid particles during a time scale {\tau}. It is argued
that these two increments probe preferentially the stretching and spinning of
material fluid elements, respectively. This property is confirmed (in the limit
of vanishing {\tau}) by examining the variances of these increments conditioned
on the local topology of the flow. Interestingly, these longitudinal and
transverse Lagrangian increments are found to share some qualitative features
with their Eulerian counterparts. In particular, direct numerical simulations
at turbulent Reynolds number up to 300 show that the distributions of the
longitudinal increment are negatively skewed at all {\tau}, which is a
signature of time irreversibility of turbulence in the Lagrangian framework.
Transverse increments are found more intermittent than longitudinal increments,
as quantified by the comparison of their respective flatnesses and scaling
laws. Although different in nature, standard Lagrangian increments (projected
on fixed axis) exhibit scaling properties that are very close to transverse
Lagrangian increments.
