Radio button uncheck on second click

I'll suggest to add a custom attribute to keep track of each radio's previous state like so:

$(function(){
    $('input[name="rad"]').click(function(){
        var $radio = $(this);

        // if this was previously checked
        if ($radio.data('waschecked') == true)
        {
            $radio.prop('checked', false);
            $radio.data('waschecked', false);
        }
        else
            $radio.data('waschecked', true);

        // remove was checked from other radios
        $radio.siblings('input[name="rad"]').data('waschecked', false);
    });
});

You will also need to add this attribute to the initially checked radio markup

<input type="radio" name="rad" id="Radio0" checked="checked" data-waschecked="true" />

JSFIDDLE DEMO

UPDATE:

 $(function(){
        $('input[name="rad"]').click(function(){
            var $radio = $(this);

            // if this was previously checked
            if ($radio.data('waschecked') == true)
            {
                $radio.prop('checked', false);
                $radio.data('waschecked', false);
            }
            else
                $radio.data('waschecked', true);

            // remove was checked from other radios
            $radio.siblings('input[type="radio"]').data('waschecked', false);
        });
    });

But do ensure that you don't have other radio-groups to work with, otherwise you have to provide some attributes to specify these buttons like name prop I focused earlier.

This should work more generally than the other solutions as it handles cases where all radios are not in the same parent.

var $radios = $('input[type="radio"]');
$radios.click(function () {
  var $this = $(this);
  if ($this.data('checked')) {
    this.checked = false;
  }
  var $otherRadios = $radios.not($this).filter('[name="'
                                               + $this.attr('name') + '"]');
  $otherRadios.prop('checked', false).data('checked', false);
  $this.data('checked', this.checked);
});