Sedimentary layers affected by vertical compaction and strong lateral tectonic stresses are often characterized by low anisotropic symmetry (e.g., tilted orthorhombic [TOR]/monoclinic or even triclinic). Considering all types of pure-mode and converted waves, we derive the normal moveout (NMO) series coefficients of near normal-incidence reflected waves in arbitrarily anisotropic horizontally layered media, for a leading error term of order six. The NMO series can be either a function of the invariant horizontal slowness (slowness domain) or the surface offset (offset domain). The NMO series coefficients, referred to also as effective parameters, are associated with the corresponding azimuthally varying NMO velocity functions. We distinguish between local (single-layer) and global (overburden multilayer) effective parameters, which are related by forward and inverse Dix-type transforms. We derive the local effective parameters for an arbitrary anisotropic (triclinic) layer, which is the main contribution of this paper. With some additional geologic constraints, the local effective parameters can then be converted into the interval elastic properties. To demonstrate the applicability of our method, we consider a synthetic layered model in which each layer is characterized with TOR symmetry. The corresponding global effective model loses the symmetries of the individual layers and is characterized by triclinic symmetry.