In all recent near-optimal sorting algorithms for meshes, the packets are sorted with respect to some snake-like indexing. In this paper we present deterministic algorithms for sorting with respect to the more natural row-major indexing. For 1-1 sorting on an n × n mesh, we give an algorithm that runs in 2 · n+o(n) steps, matching the distance bound, with maximal queue size five. It is considerably simpler than earlier algorithms. Another algorithm performs k-k sorting in k · n/2+o(k · n) steps, matching the bisection bound. Furthermore, we present uniaxial algorithms for row-major sorting. Uni-axial algorithms have clear practical and theoretical advantages over bi-axial algorithms. We show that 1-1 sorting can be performed in 2 1/2 · n+o(n) steps. Alternatively, this problem is solved with maximal queue size five in 4 1/3 · n steps, without any additional terms. For practically important values of n, this algorithm is much faster than any algorithm with good asymptotical performance. A hot-potato sorting algorithm runs in 5 1/2 · n steps.