Hierarchial clustering cannonnical representation?
You can always adjust your algorithm to use a deterministic tie-breaking rule, e.g., sort the data points before applying the clustering algorithm, and always break ties in favor of the earliest/leftmost item.  This yields a deterministic algorithm, so if you give it exactly the same input both times, it will produce the same clustering both times.  I can't tell whether this is what you are looking for or not.