American here: we have a big “this side faces street” on the side opposite the handle. We also have trucks with hydraulic arms that lift the bins up, so the two things might be related: the bin has to be in a certain orientation so the arm can grab it easily.
You could always call the service and ask. I’m guessing they don’t care about which way the handle faces as much as they care about things like overfilling or lots of liquid in the bin.