This paper concentrates on a class of pseudorandom sequences generated by combining q-ary m-sequences and quadratic characters over a finite field of odd order, called binary generalized NTU sequences. It is shown that the relationship among the sub-sequences of binary generalized NTU sequences can be formulated as combinatorial structures called Hadamard designs. As a consequence, the combinatorial structures generalize the group structure discovered by Kodera et al. (IEICE Trans. Fundamentals, vol.E102-A, no.12, pp.1659-1667, 2019) and lead to a finite-geometric explanation for the investigated group structure.