  Two vector-like families 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 in the context of a viable and economical SUSY composite model. One of
the many attractive features of the model is an explanation of the inter-family
mass-hierarchy for which the existence of the two vector-families is crucial.
This paper is devoted to a detailed study of the expected masses, mixings and
decay modes of the fermions belonging to the two vector-like families. The
masses of the vector-like quarks are expected to lie in the range of 500 GeV to
about 2.5 TeV, while those of the vector-like 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 LHC
and, possibly a future version of the SSC are built, pair-production of the
vector-like quarks would lead to systems such as ($b\overline{b}+4Z+W^+W^-$)
and ($b\overline{b}+2Z+W^+W^-$), while an $e^-e^+$ linear collider (NLC) of
suitable energy can produce appreciably a single neutral heavy lepton $N$
together with $\nu_\tau$, followed by the decay of $N$ into $(Z+\nu_\tau)
\rightarrow (e^-e^+) + \nu_\tau$. This last signal may conceivably materialize
even at LEP 200 if $N$ is lighter than about 190 GeV.
