The possibility is investigated that some physical mesons contain appreciable components both of qq-bar (quark-antiquark) states and of gluon bound states. Experimental data suggest this type of mixing both for pseudoscalar mesons and scalar mesons. Since the interactions of gluon bound states with qq-bar states are not known accurately theoretically, no definitive test of this type of mixing can be given at present. However, a consistency test for gluon-bound-state mixing is given and shown to be satisfied by a recent analysis of the pseudoscalar-meson interactions. An extension of this analysis suggests that the eta (549) has an appreciable gluon-bound-state component while the eta' (958) does not. The scalar-meson data are anomalous unless there is appreciable mixing of the lightest qq-bar states with some other kind of state. The masses of the light scalar and pseudoscalar gluon bound states that are suggested by the data are in the regions expected. Some differences between the qq-bar and qqq-barq-bar models of the scalar mesons are discussed. Better data concerning scalar mesons are needed, but some significant conclusions may be drawn from the limited data available at present.