For some reason I have it in my head that the "up" and "down" directions reverse at the Scotland/England border (e.g. if you're at Reston then "up" is towards Edinburgh, but if you're at Chathill, "up" is towards London) but googling it I can't find any mention of this. Have I imagined it?
As mentioned, this isn't correct. What you may be thinking of is that the mileposts north of the border are from a zero at Edinburgh, so start decreasing when crossing the border in either direction. Most often the mileage increases in the Down direction, but not always.
For a lot of, if not all former Midland Railway routes, the centre mileage point is Derby. For me in Selly Oak for example, up is northbound.
This is true, but it isn't because the Midland decided to take its mileage from Derby. All Midland routes were re-numbered in the early 20th century starting from a zero at St Pancras. Where the junction for a branch was trailing when travelling from St Pancras, that branch had its own zero at the junction and mileage increased going away from it. The line from Derby all the way to Bristol was treated in exactly the same way as any other trailing branch.
Historically it was always 'Up' towards the railway companies' head quarters.
Not always, see above. The Midland was Down from St Pancras to Derby and the Scottish lines were Down from the English border. Generally an attempt was probably made to minimise the flipping of Up and Down for the principal routes, which avoids confusion when many
trains are also described as up or down. This may or may not explain why the L&Y was Down in all directions from Victoria - this was their HQ but it also happened to avoid up/down reversal for the main routes at junctions with other companies such as Stalybridge, Wigan and Euxton.
Certainly France uses pair and impair - even and odd. Even is towards Paris - nicely starting with P.
Italy uses pari and dispari - again even and odd. However, in Italy, pari is North or West-bound, impari is South or East. Ironically, Parigi is generally North West of Italy...
You'll probably find that the signal numbers are also even and odd for the appropriate directions. More recent British signalling schemes do this too with even numbers in the Up direction, but many older ones don't.