Two vectorlike familes QL,R=(U, D, N, E)L,R and QL,R′;=(U′;, D′;, N′;, E′;)L,R with masses of order 1 TeV, one of which is a doublet of SU(2)L and the other a doublet of SU(2)R, have been predicted to exist, together with the three observed chiral families, in the context of a viable and economical SUSY preon model. The model itself possesses many attractive features which include explanations of the origins of (i) diverse mass scales, (ii) family replication, (iii) protection of the masses of the composite quarks and leptons compared to their compositeness scales, and (iv) interfamily mass hierarchy. The existence of the two vectorlike familes, a prediction of the model, turns out to be crucial especially for an explanation of the interfamily mass hierarchy (IFMH). Given the simplicity of the explanation, the observed IFMH in turn appears to us to be a strong hint in favor of the existence of the two vectorlike families. This paper is devoted to a detailed study of the expected masses, mixings and decay modes of the fermions belonging to the two vectorlike families, in the context of the SUSY preon model, with the inclusion of the renormalization effects due to the standard model gauge interactions. Including QCQ renormalization effects, the masses of the vectorlike quarks are expected to lie in the range of 500 GeV to about 2.5 TeV, while those of the vectorlike leptons are expected to be in the range of 200 GeV to 1 TeV. Their mass pattern and decay modes exhibit certain distinguishing features and characteristic signals. For example, when the CERN LHC and, possibly a future version of the SSC are built, pair production of the vectorlike quarks would lead to systems such as (bb̄+4Z+W+W-) and (bb̄+2Z+W+W-), while an e-e+ linear collider (NLC) of suitable energy can produce appreciably a single neutral heavy lepton N together, with ντ, followed by the decay of N into (Z+ντ)→(e-e+)+ντ. This last signal may conceivably materialize even at CERN LEP 200 if N is lighter than about 190 GeV. © 1995 The American Physical Society.