There are many different versions of DTW. Most often, different versions of DTW use a different step pattern (where can the sequences move with respect to each other, and what is the cost of this). In most cases a symmetric step pattern is used, and then DTW distance will be symmetric. However, if an asymmetric step pattern is used, DTW distance can be asymmetric as well.
However: Even with a symmetric step pattern (even with the default pattern) DTW distance is not a metric. It is easy to produce some examples where the triangle inequality is violated. So it is never easy to interpret DTW distance in terms of a metric.
How would you interpret an asymmetric DTW distance? If you are not using an asymmetric step pattern, this will not happen, so there is no special interpretation. If you use an asymmetric step pattern, there probably is a reason for deviating from the default pattern. So the interpretation depends on that reason as well.
From the reference you posted: "Several step patterns have been discussed in the literature. A classic paper by Sakoe and Chiba
(1978) classifies them according to two properties: their symmetry (symmetric/asymmetric),
and the bounds imposed on the slope expressed through a parameter P. The eight step
patterns shown in Sakoe and Chiba (1978, Table I) are pre-defined in dtw, with names
symmetricP1, asymmetricP05, and so on.1 All of them are normalizable."