Hello Al,

Thank you for using our community.

The Default styles for the WebDatePicker comes with a calendar icon as a drop-down button. If you want the drop-down icon you can use the CustomButton-ImageUrl property and choose the desired image. Here is a code sample:
        <ig:WebDatePicker ID="WebDatePicker1" runat="server" Buttons-CustomButton-ImageUrl="~/ig_res/Default/images/igdd_DropDownButton.png"></ig:WebDatePicker>

If you have further questions, don’t hesitate to contact me.