19. (15 points) As shown in Figure 4, there is a pointer on a disk, initially pointing to the top of the disk. The pointer rotates clockwise around the center of the disk by an angle $\alpha$ each time, and $3.6^{\circ}<\alpha<180^{\circ}$. After 2,004 rotations, it returns to its initial position for the first time, pointing to the top of the disk again. How many different possible values can $\alpha$ have?
19. Clearly, if
$3.6^{\circ}1$, then let
$n_{1}=\frac{n}{d}, n_{2}=\frac{2004}{d}$.
At this point, we have $\alpha=\frac{n_{1} \times 360^{\circ}}{n_{2}}$.
This means that after the pointer rotates $n_{2}$ times, each time rotating by $\alpha$, the pointer will rotate $n_{1}$ full circles and return to its initial position, which contradicts the problem statement.

From the above discussion, we know that for any $n$ satisfying $21 \leqslant n \leqslant 1001$ and $(n, 2004)=1$, there corresponds a possible $\alpha$. Conversely, this is also true. Therefore, the problem becomes finding the number of all $n$ that satisfy the above two conditions.
Since $2004=2^{2} \times 3 \times 167$,
we have $(n, 2004)=1 \Leftrightarrow 2 \chi_{n}, 3 \psi_{n}, 167 \uparrow_{n}$.
Among the positive integers not greater than 1001, the number of positive integers that are not divisible by 2 or 3 is
$$
\begin{array}{l}
1001-\left(\left[\frac{1001}{2}\right]+\left[\frac{1001}{3}\right]-\left[\frac{1001}{2 \times 3}\right]\right) \\
=1001-(500+333-166)=334 \text { (numbers). }
\end{array}
$$
(The symbol $[a]$ represents the greatest integer not exceeding $a$.)
Among them, only $1 \times 167$ and $5 \times 167$ are divisible by 167, so the number of $n$ not greater than 1001 and satisfying the conditions is $334-2=332$. Removing the 7 numbers $1, 5, 7, 11, 13, 17, 19$ that are not greater than 20, we find that the number of $n$ satisfying both conditions is $332-7=325$.
Therefore, $\alpha$ has 325 possible distinct values.