In this work, the concept of high-order space harmonics (HSHs) applied to a hybrid form of nonradiative dielectric (NRD) waveguide and substrate-integrated square waveguide (SIW) involving both TE10 and TE01 modes is investigated with aim to develop multi-functional dually polarized leaky-wave antennas (DP-LWAs). A polarization-selective coupling (PSC) mechanism is used to guide the two orthogonally oriented modes in the hybrid structure to achieve a polarization diversity feature. The polarization effect on unit-cell geometrical structures is examined for different characteristics of LWA such as propagation constant behavior, leakage ratio, and scanning range variation. In addition, we demonstrate how the extrinsic characteristics of HSHs in a one-dimensional periodic LWA and its unit-cell analysis can provide design flexibility through two case studies. In the first case of demonstration, we show a dually polarized multi-beam LWA design with wide scanning capability in which we use both even and odd HSHs of n-1,-2). The second antenna design case is concerned with a DP-LWA having the capability of fixed and scanning beams operation by using a combinatory set of PSC and four HSHs of n-1,-2,-3,-4. Two DP-LWA prototypes are demonstrated as proofs of concept, and the obtained experimental results show a very good agreement with their simulation and analysis counterparts.