Got a little further, so updating:
Using bootstrap as UI, have a group of radio buttons that are styled by bootstrap as such:
<div id="radio-group-follow" class="btn-group" data-toggle="buttons">
<asp:RadioButton GroupName="myGroup1" OnCheckedChanged="Radio1_CheckedChanged"
AutoPostBack="true" ID="Radio1" runat="server" Checked="False"
CssClass="btn btn-default" Text="1"></asp:RadioButton>
</div>
So normally, the above code should trigger postback on its own. Bootstrap wraps the buttons with a DIV which blocks the postback from happening.
The following JS will fire 1 post back, but only works once...
$(function () {
$("#<%=Radio1.ClientID%>").change(function (event) {
__doPostBack('Radio1', '');
});