  We extend the block filtration, defined by Brown based on the work of
Charlton, to all motivic multiple zeta values, and study relations compatible
with this filtration. We construct a Lie algebra describing relations among
motivic multiple zeta values modulo terms of lower block degree, proving
Charlton's cyclic insertion conjecture in this structure, and showing the
existence of a `block shuffle' relation, and a previously unknown dihedral
symmetry and differential relation.
