What methods are available to evaluate similarity between different clustering algorithms? I am performing extensive customer segmentation analysis and so far implemented Gaussian Mixture Models, K-Means, and Hierarchical Clustering. For the most part, the algorithms agree on the structure of the clusters and well as the number (7-8). I would like to know if there is a common method to either...


compare similarity between clusters. Can you apply Adjusted Rand Index to two different clusterings of the same data (k-means clusters vs gmm)? I was under the impression ARI is used in instances where you know the truth of the data.
Find the common clusters within the clusterings. If all of the algorithms say one cluster is defined by high spending, then is there a way to determine the best centroid(s) to use for a "Master" cluster? Is it common to cluster the cluster results?
If you just want to see how similar the clustering is between 2 algorithms, using the sklearn.metrics.adjusted_rand_score() function is a good starting point. This will work for unsupervised learning, no need for a label.

https://scikit-learn.org/stable/modules/generated/sklearn.metrics.adjusted_rand_score.html

Or are you looking at choosing the best combined grouping overall? I think "clustering the cluster" results is not a common approach, however there are a few articles online on how this could be done. But I don't know of any packages that will do this for you. 
One idea is to create a new column for each algorithm with the grouping chosen and then do some comparisions or calculations on these columns to get a final composite clustering.