It is really just a matter of where the motion comes from. Let me explain with an analogy.
Imagine holding a rope at one end, which is fixed at the other end to some mechanism that can measure energy transported by the wave on the rope. You starts shaking the rope, and then energy start flowing from you to the other end of the rope; as you can guess, it is not really a matter of left to right or vice-versa. The energy flow is from its source (in my example, you shaking the rope) through the medium - the rope (in every possible direction).
If you held the rope at its center and shook it, energy (and thus the wave) would propagate in both directions.
In water waves (for example the one you see after throwing a pebble in a lake), the energy moves from the pebble outwards, forming circles.
You can also see that, in your example, since every particle is moving up and down in harmonic motion, it doesn't really matters if you start by moving them up or down, and it wouldn't change the direction of energy flow.