Updated 2018
For the original question based on Bootstrap 3, the solution was to use push-pull.
In Bootstrap 4 it's now possible to change the order, even when the columns are full-width stacked vertically, thanks to Bootstrap 4 flexbox. OFC, the push pull method will still work, but now there are other ways to change column order in Bootstrap 4, making it possible to re-order full-width columns.
Method 1 - Use flex-column-reverse
for xs
screens:
<div class="row flex-column-reverse flex-md-row">
<div class="col-md-3">
sidebar
</div>
<div class="col-md-9">
main
</div>
</div>
Method 2 - Use order-first
for xs
screens:
<div class="row">
<div class="col-md-3">
sidebar
</div>
<div class="col-md-9 order-first order-md-last">
main
</div>
</div>
Bootstrap 4(alpha 6): http://www.codeply.com/go/bBMOsvtJhD
Bootstrap 4.1: https://www.codeply.com/go/e0v77yGtcr
Original 3.x Answer
For the original question based on Bootstrap 3, the solution was to use push-pull for the larger widths, and then the columns will show is their natural order on smaller (xs) widths. (A-B reverse to B-A).
<div class="container">
<div class="row">
<div class="col-md-9 col-md-push-3">
main
</div>
<div class="col-md-3 col-md-pull-9">
sidebar
</div>
</div>
</div>
Bootstrap 3: http://www.codeply.com/go/wgzJXs3gel
@emre stated, "You cannot change the order of columns in smaller screens but you can do that in large screens". However, this should be clarified to state: "You cannot change the order of full-width "stacked" columns.." in Bootstrap 3.