The concepts associated with chirality and helicity are not really opposites or complimentary. While helicity is something that one can quantify for a particular state or field, chirality is just a property that something can have.
So, to be more explicit, helicity is defined as the angular momentum component in the direction of the momentum. One can define an operator for the helicity of a state by projecting the angular momentum operator on the unit vector in the direction of the momentum.
On the other hand, chirality is the property where something has a kind of twist associated with it, such that it does not look like its mirror image. This gives rise to the concept of handedness, so that one can distinguish between right-handed and left-handed objects/entities.
In particle physics, one finds chirality in the sense that many fields, particularly the fermion fields, are distinguished between the left-handed and right-handed fields. One can, for instance, get the situation where theories have a chiral symmetry, which means that the theory is unchanged when the left-handed and right-handed fields are transformed independently. Parity, on the other hand, is a symmetry where a theory is unchanged when the left-handed and right-handed fields are interchanged.