*The notion of bi-inner product functionals*
$$P(f,g) = \sum\limits_n{<f,f_n>
<g,g_n>}$$
*generated by two Bessel sequences* {*f*_{n}} *and* {*g*_{n}} *of functions from L*^{2}*was introduced in our earlier work as a vehicle to identify dual frames and bi-orthogonal Riesz bases of L*^{2}. *The objective was to find conditions under which P is a constant mu<iple of the inner product* <*f,g*> *of L*^{2}. *A necessary and suffici condition derived in is that P is both spatial shift-invariant and phase shift-invariant. A<hough these two shift-invariance properties are, in general, unrelated, it could happen that one is a consequence of the other for certain clases of Bessel sequences* {*f*_{n}} *and* {*g*_{n}}. *In this paper, we show that, indeed, for localized cosines with two-overlapping windoes* (*i.e., only adjacent window functions are allowed to overlap*), *spatial shift-invariance of P is already sufficient to guarantee that P is a constant mu<iple of the inner product, while phase shift-invariance is not. Hence, phase shift-invariance of P for two-overlapping localized cosine Bessel sequences is a consequence of spatial shift-invariance, but the converse is not valid. As an application, we also show that two families of localized cosines with uniformly bounded and two-overlapping windows are bi-orthogonal Riesz bases of L*^{2}, *if and only if P is spatial shift-invariant. In addition, we apply this resu< to generalize a resu< on characterization of dual localized cosine bases in our earlier work in to the mu<ivariate setting. A method for computing the dual windows is also given in this paper.*