Solution:

$4004$. There are $2002$ possible positions for the $1$. The two numbers adjacent to the $1$ must be $2$ and $3$; there are two possible ways of placing these. The positions of these numbers uniquely determine the rest: for example, if $3$ lies clockwise from $1$, then the number lying counterclockwise from $2$ must be $4$; the number lying clockwise from $3$ must be $5$; the number lying counterclockwise from $4$ must now be $6$; and so forth. Eventually, $2002$ is placed adjacent to $2000$ and $2001$, so we do get a valid configuration. Thus there are $2002 \cdot 2$ possible arrangements.