Just to add to what BigT said, there is a diurnal variation to body temperature which persists even when someone has a fever. In other words, there is a natural cycling of body temperature with the highest temperature occurring late in the afternoon and persisting to early AM. Body temperature starts to drop later in the morning, say around 04:00 (give or take). The drop corresponds to a rise in the level of blood cortisol, a hormone with many properties including lowering of body temperature. Levels of cortisol peak around 08:00.
In terms of fever causing you to feel hot/cold, there is a natural sequence governing fever. Specifically, as your temperature rises, you feel cold and may even start to shake or have rigors (this is a manifestation of your body generating heat, just as would happen if you were stuck outside on a freezing day - you’d start to shake).
When the body (and specifically the hypothalamus) no longer wants to keep the body temperature so high, signals are sent to dissipate heat. Again, just as would happen if you were sweltering on a hot day, you begin to sweat and get flushed (both of which cause heat to be lost from the body).
To a large extent, then, if someone is having shaking chills (rigors), or is having drenching sweats (especially at night), it’s usually tantamount to them declaring that they are having fevers. This is often seen in patients with certain cancers or chronic infections even if they aren’t aware of, or don’t think they’re having a fever.