We are investigating a new class of linear characteristics schemes along finite-length tracks for solving the transport equation for neutral particles with scattering anisotropy. These algorithms are based on diamond differencing, as implemented with the method of discrete ordinates. The quadratic-order diamond-differencing (DD1) scheme is based on linear discontinuous coefficients that are derived through the application of approximations describing the mesh-averaged spatial flux moments in terms of spatial source moments and of the beginning- and end-of-segment flux values. This DD1 linear characteristics scheme is inherently conservative. This approach is an improvement relative to other linear characteristics schemes because no information needs to be collected on internal surfaces. Consequently, the DD1 scheme is compatible with existing tracking files for the collision-probability method. The proposed scheme is verified in one-dimensional slab geometry where it is found to be equivalent to a discrete ordinates solution and on simple two-dimensional benchmarks made of regular squares or hexagons.