http://resources.arcgis.com/en/help/arcgis-rest-api/index.html#/Query_Feature_Service_Layer/02r3000000r1000000/
Query (Feature Service/Layer)
Request parameter: returnCentroid
That's a good question. From the "What's New", it looks like it didn't appear until 10.6
What's new in the ArcGIS REST API—ArcGIS REST API: Services Directory | ArcGIS for Developers
... but since the OP was from 2016, I seems like it would have been earlier like 10.5.x....?
Dan
We too are still on ArcGIS Server 10.4 and had to come up with a solution. Perhaps not the most performance-optimized, we opted to create a publish a Geoprocessing service to return the centroid coordinate information for other map services. Basically, the GP service is a proxy or a wrapper for the desired polygon map service which we'd like centroid coordinates and some very simple python delivers that info.
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