The complex scattering parameters (S11* and S21*), relative dielectric permittivity (er* = er' - jer"), relative magnetic permeability (mr* = mr' - jmr") and absorption characteristics of some thermoplastic natural rubber (TPNR) – Fe3O4/YIG composites were investigated by means of a microwave vector network analyser and specular absorber method. The measurements were performed in the frequency range of 1 – 13 GHz with and without the presence of a transverse external magnetic field with magnitude of 1 T. The specular absorber method is used in determining the reflection loss (RL) of the composites and its dependence on material properties, thickness and the external magnetic field. In toroidal form, the composites under study seem to transmit more (Pt > 60%) but absorb and reflect less (Pa + Pr < 30%) microwave power in the frequency range used in this study and in both unmagnetised and magnetised states. The external magnetic field is seen to have the effects of reducing S11*, S21*, er', er", mr' and mr". The suppression of the relaxation and resonance behaviours on the mr' and mr" plots for all samples are observed for measurements performed in the presence of the external field. The composites, as revealed by the specular absorber method, show conditions of minimal reflection on RL versus frequency plot for all sample thickness where more than 95% of the microwave power is not reflected back. The location of the dips of minimal reflection on the frequency domain depend on the thickness, the dielectric and magnetic properties of the materials. The conditions of minimal reflection was found to occur when the thickness (t) of the sample equals the odd number multiple of a quarter wavelength in the material (lm), t = nlm/4 (n = 1, 3, 5, 7 …), where a geometrical cancellation took place at the surface of the absorber between the reflected waves, resembling a thin filem inteference phenomenon. The dips are suppressed and shifted to a higher frequency in the presence of the external magnetic field. With the knowledge of the dielectric permittivity and magnetic permeability of a material, the specular absorber method may provides a simple theoretical graphic aids in determining the absorption characteristics and location of the matching condition in the frequency domain.