Starting with the mobile version first, you can achieve what you want, most of the time.
Examples here:
http://jsbin.com/wulexiq/edit?html,css,output
<div class="container">
<h1>PUSH - PULL Bootstrap demo</h1>
<h2>Version 1:</h2>
<div class="row">
<div class="col-xs-12 col-sm-5 col-sm-push-3 green">
IN MIDDLE ON SMALL/MEDIUM/LARGE SCREEN
<hr> TOP ROW XS-SMALL SCREEN
</div>
<div class="col-xs-12 col-sm-4 col-sm-push-3 gold">
TO THE RIGHT ON SMALL/MEDIUM/LARGE SCREEN
<hr> MIDDLE ROW ON XS-SMALL
</div>
<div class="col-xs-12 col-sm-3 col-sm-pull-9 red">
TO THE LEFT ON SMALL/MEDIUM/LARGE SCREEN
<hr> BOTTOM ROW ON XS-SMALL
</div>
</div>
<h2>Version 2:</h2>
<div class="row">
<div class="col-xs-12 col-sm-4 col-sm-push-8 yellow">
TO THE RIGHT ON SMALL/MEDIUM/LARGE SCREEN
<hr> TOP ROW ON XS-SMALL
</div>
<div class="col-xs-12 col-sm-4 col-sm-pull-4 blue">
TO THE LEFT ON SMALL/MEDIUM/LARGE SCREEN
<hr> MIDDLE ROW XS-SMALL SCREEN
</div>
<div class="col-xs-12 col-sm-4 col-sm-pull-4 pink">
IN MIDDLE ON SMALL/MEDIUM/LARGE SCREEN
<hr> BOTTOM ROW ON XS-SMALL
</div>
</div>
<h2>Version 3:</h2>
<div class="row">
<div class="col-xs-12 col-sm-5 cyan">
TO THE LEFT ON SMALL/MEDIUM/LARGE SCREEN TOP ROW ON XS-SMALL
</div>
<div class="col-xs-12 col-sm-3 col-sm-push-4 orange">
TO THE RIGHT ON SMALL/MEDIUM/LARGE SCREEN
<hr> MIDDLE ROW ON XS-SMALL
</div>
<div class="col-xs-12 col-sm-4 col-sm-pull-3 brown">
IN THE MIDDLE ON SMALL/MEDIUM/LARGE SCREEN
<hr> BOTTOM ROW XS-SMALL SCREEN
</div>
</div>
<h2>Version 4:</h2>
<div class="row">
<div class="col-xs-12 col-sm-4 col-sm-push-8 darkblue">
TO THE RIGHT ON SMALL/MEDIUM/LARGE SCREEN
<hr> TOP ROW XS-SMALL SCREEN
</div>
<div class="col-xs-12 col-sm-4 beige">
MIDDLE ON SMALL/MEDIUM/LARGE SCREEN
<hr> MIDDLE ROW ON XS-SMALL
</div>
<div class="col-xs-12 col-sm-4 col-sm-pull-8 silver">
TO THE LEFT ON SMALL/MEDIUM/LARGE SCREEN
<hr> BOTTOM ROW ON XS-SMALL
</div>
</div>
</div>