The theory behind the quarterwave vertical is the monopole above a ground
plane, where the ground plane reflects the wave emitted by the vertical. The
monopole is explained using image theory. In practice, the ground plane is
replaced by radials. Do the radials reflect
the wave then?

The reflecting element on a Yagi manages to reflect most of the wave. The
reflecting element on a Yagi is a parasitic element that has an impedance to
cause the wave emitted by the driven element to flow in a particular
direction. A Yagi normally has only one reflector. Although the reflector is
in the near field of the Yagi, can a comparison be made with the radials of
a quarterwave vertical antenna? The reflector on a Yagi is usually a thin
tube with lots of air (gap) around it. Even though it occupies a small area,
it still manages to reflect most of the wave. Yagi has a Front to Back ratio
in dB.

Radials can be tuned. Some antennas have loading coils in the radials.

I notice that RF experts cannot agree on whether the radials reflect the
wave or not.

Antenna theory is often about wires and metallic items reflecting waves, and
the phase of the reflected wave. The phase of the reflected wave can be
constructive or destructive, affecting the impedance of the antenna. If an
antenna is mounted too close to the ground, the reflected wave cancels out
the emitted wave.

Because a ground plane reflects the wave, the impedance of an antenna can
vary with height.

Parastic elements on a Yagi have a mutual impedance to each other. Would you
regard the radials on a quarterwave vertical as having a mutual impedance?

The radials increase the conductivity below the radiating element,
decreasing ground losses. The radials are regarded as a finite or imperfect
ground plane.

References:
"Antenna Theory and Design" by Warren Stutzman and Gary Thiele. pages 66 to
68. Practical monopole with radial wires to simulate a ground plane.
"Antenna Engineering Handbook" by Richard C. Johnson. Radials suppress
currents from flowing on outside of coax. p 28. If the ground is imperfect,
the perfect reflected image is mutiplied by a complex ground reflection
coefficient. The ground has a mutual impedance.
"Antenna Theory" by Professor Constantine Balanis. Second Edition p 165. A
ground plane formed by a perfect conductor completely reflects the wave. If
the ground is finite i.e. not as conductive, it still reflects the wave but
not as well. The conductivity determines the quality of the reflection.