The text below applies to vertical quarterwave antenna in free space with
four quarterwave radials.

In transmit mode, the field from the radials cancel out.

When the antenna is receiving, the incoming wave interacts with the radials
and is reflected into the vertical element.

How does this fit in with the antenna reciprocity theorem? The antenna field
arrangement is working in different ways between transmit and receive. In
transmit mode, the radials cancel out, and in receive mode the radials
reflect the wave to develop a signal in the vertical element.

In receive mode, the radials strengthen the signal in the vertical element
by reflecting RF energy into the vertical element and also providing the
other half of the antenna by providing a RF ground side. The signal exists
as a potential difference between two conductors.

In transmit mode, the radials have a standing wave on them and provide the
RF ground side by 'converting' the high voltage at open circuit end to a low
voltage (RF ground with high current) at base of antenna.

Do the radials have a standing wave on them in receive mode? Does antenna
reciprocity theorem agree that the antenna can form same feedpoint impedance
and field pattern in both transmit mode and receive mode, even though the
electromagnetic field works differently? Are the electromagnetic field
patterns noticeably different according to whether antenna is receiving or
transmitting?

The vertical element is always known as the radiating element.