But I've never seen that happen.
You haven't looked then.
The rising or setting Moon is rather reddish, just as is the rising or setting Sun. However, there is a difference between the Moon and the Sun. You can look directly at the Moon, even a full Moon, regardless of where it is in the sky. On the other hand, you can only glance at the Sun when it is rising or setting. You cannot even glance at the Sun if it is at any appreciable distance above the horizon.
Both the Sun and the Moon appear reddish when they are very close to the horizon. This reddening diminishes rapidly with increasing angle above the horizon. You don't see this with the Sun because you cannot look at the Sun except when it is extremely close to the horizon. People perceive the Sun as yellow or red because the only time you can look at the Sun is when it is at sunrise or sunset. During most of the day, sunlight is white (as perceived by the human eye).
On the other hand, you can look directly at the Moon, regardless of its elevation. The rising Moon appears to be quite red on the horizon. This quickly changes to yellow and then to white as the elevation angle increases. You would see the same phenomenon if you could look directly at the Sun. Your camera can look directly at the Sun. The color of the rising Sun changes from red to yellow to white quite rapidly.