Negasum is the operation taking (x,y) to -(x+y).
Lemma: A set of cards is closed under negasum if and only if it is closed under affine combinations (i.e. it is an affine subspace).
Proof sketch: The "if" is obvious, as negasum is an affine combination.
For the "only if", proceed by induction on the number of terms in the affine combination. The base case, with only a single term, is clear.
If there is a term with a coefficient 2, rearrange it to the front to get the form 2x + sum(a_i y_i), which can be rewritten as -(x + sum((-a_i)y_i)). The inner sum is an affine combination with fewer terms, since sum(-a_i) = -sum(a_i) = -(1 - 2) = 1.
Otherwise, there is no term with coefficient 2, so the sum has the form x + z + sum(a_i y_i), which can be rewritten as -(-(x+z) + sum(-(a_i)y_i)), where again the inner sum is an affine combination with fewer terms.