I'm having trouble understanding this, because under your definition, T(-z) requires T to be defined on the lower half-plane.  The series doesn't converge there.

Also, N(v) should be half of the squared norm.  Equivalently, N(v) should be the value of the quadratic form that defines the inner product on the lattice.

I can think of one interpretation, where the theta function is actually a Jacobi form, defined as a section of a line bundle on the universal elliptic curve.  This can be viewed as a function T(t,z) on HxC, with some invariance properties under translation by lattice elements in C and SL(2,Z) transformations in H.  Then you can negate the z variable.

Eichler and Zagier have a book on Jacobi forms, called <i>The theory of Jacobi forms</i>.