I figured out my own answer.  I found that there are TWO DualList controls.  One is an ASPX based WebControl which does not function with POSTBACKS, called DualList.  The other is a class derived server-based control that dynamically draw itself, called DualListBox.

For anyone with this same problem, this is what I did:
   1) Create a new WebControl
   2) Add assembly registration to the Markup page:
          <%@ Register TagPrefix="DNN" Namespace="DotNetNuke.UI.WebControls" Assembly="DotNetNuke" %>
   3) Add the DNN DualListBox server-based control to your page:
          <DNN:DualListBox ID="dlDualList" runat="server"
                                             AddKey="Add" 
                                             RemoveKey="Remove"
                                             AddAllKey="AddAll" 
                                             RemoveAllKey="RemoveAll"
                                             AddImageURL="~/Images/rt.gif"
                                             AddAllImageURL="~/Images/ffwd.gif"
                                             RemoveImageURL="~/Images/lt.gif" 
                                             RemoveAllImageURL="~/Images/frev.gif"
                                             ContainerStyle-HorizontalAlign="Center">
             <AvailableListBoxStyle CssClass="NormalTextBox" Height="150px" Width="250px" />
             <SelectedListBoxStyle CssClass="NormalTextBox" Height="150px" Width="250px" />
          </DNN:DualListBox>
   4) In the codebehind of your new control, write the code for the 4 buttons Add, AddAll, Remove and RemoveAll
   5) Remember to reassign the lists, DataText property, DataValue property after each "move" and rebind the data
   6) The lists can not be retrieved from the DNN DualListBox control after a postback, so to work around this I saved my lists into SessionVariables
   7) Extra Note: The DataBind doesn't sort your lists, you may want to consider adding a ICompare sort class to handle sorting on the Bind event.