  Let ${\cal D}^k$ be the space of $k$-th order linear differential operators
on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural
1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal
D}^k$. (To define this family, one considers arguments of differential
operators as tensor-densities of degree $\lambda$.) In this paper we solve the
problem of isomorphism between $\Diff(\bf R)$-module structures on ${\cal D}^k$
corresponding to different values of $\lambda$. The result is as follows: for
$k=3$ $\Diff(\bf R)$-module structures on ${\cal D}^3$ are isomorphic to each
other for every values of $\lambda\not=0,\;1,\;{1\over 2},\;{1\over 2}\pm
\frac{\sqrt 21}{6}$, in this case there exists a unique (up to a constant)
intertwining operator $T:{\cal D}^3\to{\cal D}^3$. In the higher order case
$(k\geq 4)$ $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to
two different values of the degree: $\lambda$ and $\lambda^{\prime}$, are
isomorphic if and only if $\lambda+\lambda^{\prime}=1$.
