Identifying cluster centroids using ArcGIS Desktop?

ArcGIS has a few tools in the Spatial Statistics toolbox that might be useful.

Mean Center

Identifies the geographic center (or the center of concentration) for a set of features.

Median Center

Identifies the location that minimizes overall Euclidean distance to the features in a dataset.

Central Feature

Identifies the most centrally located feature in a point, line, or polygon feature class.

I think the Mean Center would be most useful here when used with the Case Field option.

To aggregate the clusters use grouping which produces a field SS_GROUP then for each unique SS_GROUP in the feature class create a convex hull then use Feature to Point to obtain the centroid of that particular cluster, with polygons you can specify inside to guarantee that banana shaped polygons produce a centriod that is inside the polygon but that wont matter too much as the convex hull cannot be banana shaped - that would be concave.