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I figured it out.
If I set it up this way, I'm able to have one input sent to two different params.
protected function defaultParamValues()
{
    return array(
        "startDate"=>"",
        "startDate2"=>"",
        "endDate"=>"",
        "endDate2"=>""
    );
}
protected function bindParamsToInputs()
{
    return array(
        "startDate"=>"startDate",
        "startDate2"=>"startDate",
        "endDate"=>"endDate",
        "endDate2"=>"endDate"
    );
}
    
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