With another clockwise fan from the same maker, I was able to reverse the direction by interchanging the yellow and black wires as some answers here indicated. The explanation as I understood is that the rewiring changes the winding with which the capacitor is in series and hence the starting direction is inverted.

In 3-phase motors, each of the three stator windings carry a current out of phase with others and the phase difference generates the rotating magnetic field required to cause the motion. With single-phase ones, a phase difference is engineered by splitting the single phase current into two stator windings and putting a capacitor in series with one of the windings so that there's a 90° phase difference between the currents in the two windings. This page on electric motors explains the concept with illuminating animations.

The following figure shows my guess, based on the above information, at the internal wiring of the clockwise spinning fan whose image is posted in the question, for clockwise and anti-clockwise rotations.

A point to note here is that single-phase AC itself produces a changing magnetic field - though a pulsating one, not a rotating one. But this pulsating field can be resolved, as per the double field revolving theory (the link has an excellent video of the workings by the way), into two revolving fields rotating in opposite direction to each other. These two fields produce an equal but opposite torque. On a static rotor, they'll cancel each other out. But an initial rotation makes torque in one direction greater than the other and starts up the fan.

This is what, I believe, happened when the OP switched red and yellow wires on his fan. The result was that the capacitor was in series with both windings => there was no phase difference in the currents in the windings. When he added a slight initial rotation, the fan continued spinning in the nudged direction.

Light dimmers are designed to drive loads that are largely resistive in nature like light bulbs. They are generally not compatible with loads that are inductive. Most, if not all, AC motors are inductive type loads.

That said, whether a dimmer switch will work safely with your fan or not depends entirely upon the type of motor on the fan.

There are some types of AC motors that can work to gain speed control via a dimmer switch type of circuit. These motors are often found in AC corded power tools such as electric drills and jig saws. Just as often these tools come with a variable speed capability via a proportional squeeze trigger. These tool speed controls work just like a common lamp dimmer. The motor type involved is generally the type with brushes.

Common low cost fans do not use brushed type motors. The usage life time would be just too short and non-brushed low power motors are often way cheaper to manufacture.

With this information in hand you may be able to evaluate your fan and decide if it is suitable for use with a dimmer. I suspect that it is likely not compatible.