Kuiper-belt objects (KBOs) are an ancient reservoir of comets beyond Neptune's orbit. Some of these objects were recently found to have the reddest optical colours in the Solar System, but the number of objects for which accurate colours were available was too small for any correlation to be discerned between colour and physical or dynamical properties, which might shed light on the origin of these objects. Here we report that all nine of the KBOs in our survey on near-circular (low-eccentricity) orbits with perihelion distances larger than 40 AU have extremely red surfaces, thereby connecting an observable property with a dynamical class. Of the objects with orbital eccentricities greater than 0.1, about half are also very red, while the rest have colours similar to the Sun, meaning that reflected sunlight is not strongly modified by the objects' surface properties. In addition, of the 13 'classical' KBOs (those with semimajor axis a 45 AU and eccentricity e < 0.15), the ten that are very red are in orbits with small angles of inclination to the ecliptic, whereas the three with solar colours are all in high-inclination orbits. We suggest that these three 'grey' classical KBOs may be part of a dynamical group that is separate from the 'red' classical KBOs.