In the case of a perfectly elastic demand curve, when price rises the consumer is said to demand 0 units of the said commodity. However, graphically, even when price falls demand falls to 0. This seems to violate the demand law. Is there an explanation for this phenomenon or is the graph supposed to be interpreted differently?
Your second sentence is false ("However, graphically, even when price falls demand falls to 0.").
Below is an example of perfectly elastic demand. At any price above \$1, quantity demanded (QD) is 0, as indicated by the vertical line segment on the left.†
But at any price below \$1, QD is not 0, but $\infty$. This is not depicted in the graph below, because we can't depict $\infty$ on a graph.
And so the law of demand is not violated: when price falls, QD does not also fall. Instead, QD rises.
(Note that in the real world, perfectly elastic demand does not exist. To use the above example, it is never the case that at a price of $1, every non-negative real number is a possible QD. As is often the case in economics, the concept of perfectly elastic demand is simply an abstract idealization that is occasionally helpful for explaining certain phenomena.)
†Note that this vertical line segment is often omitted. In my opinion, this omission is a mistake and is one of the reasons for the confusion on your part.
"Perfectly elastic demand" is not a situation associated with individual consumer demand, it is a concept related to the market demand schedule.
And then, it does not describe any rigorous aggregation of individual demands, but instead, it expresses demand as perceived by the firms and is related to firm competition, not consumer preferences:
If a firm increases its price it will lose all customers, but because they will shop elsewhere, not because their utility maximization tells them to stop consuming the product. So it does not say that "consumer demand will drop to zero" -it will do so for the specific firm, not for the market.
As for decreasing the price, the firm has no reason to do it, since it can sell all the product it can at the going price.