  We study the chiral transformation properties of all possible local
(non-derivative) interpolating field operators for baryons consisting of three
quarks with two flavors, assuming good isospin symmetry. We derive and use the
relations/identities among the baryon operators with identical quantum numbers
that follow from the combined colour, Dirac and isospin Fierz transformations.
These relations reduce the number of independent baryon operators with any
given spin and isospin. The Fierz identities also effectively restrict allowed
baryon chiral multiplets. It turns out that the chiral multiplets of the
baryons are equivalent to their Lorentz representation. For the two independent
nucleon operators the only permissible chiral multiplet is the fundamental one
$(\frac12,0)\oplus(0,\frac12)$. For the $\Delta$, admissible Lorentz
representations are $(1,\frac12)\oplus (\frac12,1)$ and
$(\frac32,0)\oplus(0,\frac32)$. In the case of the $(1,\frac12)\oplus
(\frac12,1)$ chiral multiplet the $I(J)=\frac32(\frac32)$ $\Delta$ field has
one $I(J)=\frac12(\frac32)$ chiral partner; otherwise it has none. We also
consider the Abelian ($U_A(1)$) chiral transformation properties of fields and
show that each baryon comes in two varieties: 1) with Abelian axial charge +3;
and 2) with Abelian axial charge -1. In case of the nucleon these are the two
Ioffe's fields; in case of the $\Delta$, the $(1,\frac12)\oplus (\frac12,1)$
multiplet has Abelian axial charge -1 and the $(\frac32,0)\oplus(0,\frac32)$
multiplet has Abelian axial charge +3.
