...meant riding in a carriage , sitting facing the horses: sitting backwards meant sitting with ones back to the horses ;-) Exactly like sitting with one back to the engine or facing the engine in trains nowadays.

In Mary Musgrove's case, what is happening is that Mr and Mrs Musgrove are sitting in the carriage facing the horses, the favoured position, whilst Mary, Henrietta and Louisa were crowded on the seat with their back to the horses, that is "backwards".

Look at this passagew from William Kitchiner's book, The Traveller's Oracle, describing the difficulties of travelling on a stage coach and the rush to get the best seats ;-0 That is, the ones facing forwards ......

People are generally anxious to secure front places either because they cannot , or fancy they cannot ride backwards; but if they travel at Night, the Wind and Rain while sitting in front will beat into their faces, the only remedy for which is to draw up the Glasses( carriage windows-JW) (a privilege vested by travelling etiquette in the occupiers of those places) and thus must they sit the remainder of the night in an Atmosphere too impure for any Gentleman who has not previously served an apprenticeship in the exhausted receiver of an Air Pump
Nothing occasions more severe Cold etc than the sudden exposure to the Cold Air immediately after coming out of one of these Vapour Baths.

Should a Person in Travelling for any disance and sitting backwards meet with Companions who close the Windows, and pertinaciously persist in prohibiting any importation of Oxygen;- if all arguments on the necessitiy of Ventilation are unavailing and your Lungs feel oppressed from the lack of Fresh Air- you may let your stick or your Umbrella fall (accidentally) against one of the windows: i.e. if you are of opinion that it is more advisible to give a Glazier 3 shillings to replace a pane of Glass, than it is to pay double that sum for Physic to remove a pain in your Head, which you will otherwise get by breathing fould air
Volume 1, pp 156-7.