Coin alignment in the UK has changed over the years.
In the days of hammered coins - pre-about 1656 in most cases - there was no set alignment. If you flip over a hammered coin the reverse can be in any direction compared to the obverse, mainly because they were done by hand.
When machine minting came in and it was easier to control the alignment, in Britain we went with keeping the devices the right way up if you flipped about a horizontal access. This became known as "coin alignment" and is what you have noticed on your 1806 halfpenny.
Medals were struck so that the devices remained the right way up when flipped about a vertical access, which makes sense when you think of them hanging on someone's chest and getting turned round as they move. This became known as "medal alignment".
Until the early 19th century all British coins were struck "coin aligned". Over a number of years this changed and by Victoria's time all were being struck "medal aligned", and are still done so today. Other countries of the world have had different ideas at different times, though almost all are now struck medal aligned.
There are some valuable rarities where a coin should be struck one way but has been mistakenly struck the other. In books you will see pairs of arrows to indicate how they should be - two arrows up for medal, one arrow each way for coin aligned.
I hope that helps!