To reproduce:

this.radDropDownList1.DropDownListElement.TextBox.ShowClearButton = true;

The clear button doesn't have image or text and it seems like it is not visible.

Note: this problem can be reproduced in other controls that use RadTextBoxElement, e.g. RadMaskedEditBox, RadSpinEditor etc.

Workaround:  this.radDropDownList1.DropDownListElement.TextBox.ClearButton.Text = "X";

NOTE: This issue is reproducible in all controls that host the text box.