  We regard a finite word $u=u_1u_2\cdots u_n$ up to word isomorphism as an
equivalence relation on $\{1,2,\ldots, n\}$ where $i$ is equivalent to $j$ if
and only if $x_i=x_j.$ Some finite words (in particular all binary words) are
generated by "{\it palindromic}" relations of the form $k\sim j+i-k$ for some
choice of $1\leq i\leq j\leq n$ and $k\in \{i,i+1,\ldots,j\}.$ That is to say,
some finite words $u$ are uniquely determined up to word isomorphism by the
position and length of some of its palindromic factors. In this paper we study
the function $\mu(u)$ defined as the least number of palindromic relations
required to generate $u.$ We show that every aperiodic infinite word must
contain a factor $u$ with $\mu(u)\geq 3,$ and that some infinite words $x$ have
the property that $\mu(u)\leq 3$ for each factor $u$ of $x.$ We obtain a
complete classification of such words on a binary alphabet (which includes the
well known class of Sturmian words). In contrast for the Thue-Morse word, we
show that the function $\mu$ is unbounded.
