Similar ESRI arcgis.com can anyone give pointers on how to move layer up/down? And if possible include the context menu.
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You can use the map's reorderLayer() method.
Changes the layer order in the map. Note that the first layer added is always the base layer, even if its order is changed.
The index refers to the location for placing the layer. The bottom most layer has an index of 0.
The client-side APIs treat each map service as a layer, while the map service itself may also contain layers (as defined in your MXD, for example). If you are asking about layers within an ArcGIS Server map service, it cannot be done as AGS does not support it.