This example demonstrates how to use the BarSplitButtonItem to select themes. All logic is incorporated into BarSplitItemThemeSelectorBehavior. So, to activate this functionality, assign the behavior to the item:
[XAML]<dxb:BarSplitButtonItem><dxmvvm:Interaction.Behaviors><dxb:BarSplitItemThemeSelectorBehavior/></dxmvvm:Interaction.Behaviors></dxb:BarSplitButtonItem>