I have a 2005 manual transmission Toyota Corolla. Just this winter, it's been doing this weird thing in the morning: I'll start the car and start driving it, and the engine starts warming up. Once it's warmed up (temperature gauge in the middle), I'll turn on the heater if the windshield isn't fully defrosted. However, it won't start blowing warm air right away. While it's blowing cold air, the temperature gauge will creep up until it's close to the red, where it'll remain for a few seconds. After that, I can hear something... shift? behind the dashboard, and hot air will start blowing, and the temperature gauge will start going down.
It only does this when it's cold, and it won't overheat on the rest of my commute. When I drive home at the end of the day, this doesn't happen.
To give context to what I mean by "cold", I live in northern California, so this winter the temperature range in a given day is from 30ish to 60ish degrees F (