I am interested to know if there is any religious meaning associated with Lashon HaKodesh being written from right to left as opposed to left to right
In The Great Partnership Rabbi Sacks gives a speculative explanation of the difference between right-left and left-right alphabets. The book is an attempt at describing the differences between science and religion generally, which he essentially divides into two distinct categories but which can be integrated together; hence "great partnership".
Sacks says that the left side of the brain is responsible to break information down into its constituent parts and see how they interact ('left brain thinking'). In contrast, the right side of the brain is responsible for joining events together so that they tell a story and are relevant for creativity and relationships ('right brain thinking'). The left-brain is responsible for analytical processes (writing, comprehension, computation) whereas the right brain is responsible for intuition and creativity.
Rabbi Sacks says that good examples of left- and right-brain thinking is science and religion. The mantra that he states throughout is:
"Science takes things apart to see how they work, religion puts things together to see what they mean".
Rabbi Sacks uses this thematic idea to contrast "Jerusalem" with "Athens". He notes that although the alphabets are similar (aleph, bet, gimmel versus, alpha, beta, gamma):
"...in the move to Greece the alphabet acquired letters to signal vowel sounds as well as consonants... Then, between the eighth and sixth centuries BCE, it evolved into a strange form of continuous writing, beginning from right to left, then, on reaching the end of the line, turning round and coming back, left to right, and so on. Eventually, in the fifth century, it settled into the new mode, left to right, where it has stayed ever since."
Greek philosophy (Socrates, Plato, Aristotle, etc.) "emphasised the universal over the particular, the timeless over the time-bound, the abstract over the concrete particular and the impersonal as opposed to the personal". He then goes on to state this as a basis for the fundamental differences in the philosophy of Jerusalem and Athens. The Torah tells a narrative story ("right brain"), whereas Greek philosophy tells of abstract universalism ("left brain").
"Could it be that this difference between the two cultures has something to do with the way their respective alphabets are written? Greece, at the time the alphabet was changing from right-left to left-right, became the world's first, and greatest, left-brain civilisation. It was not only a left-brain culture. There was greatness too, in the more right-brain fields of art, architecture and drama. But the conceptualisation and abstraction, the analysis of matter into its atomic parts, the Platonic devaluation of the personal and particular: all this comes with the unmistakable signature of the left hemisphere. The fact that this was hapenning at the same time, in the same place, as the emergence of the wrodls' first fully vowelled, left-right alphabet cannot be merely coincidental. The way we record and transmit information has a huge impact on cognitive styles.
The Torah narrative is in tune with right-brain thinking. Furthermore, the lack of vowels in hebrew also gives the Torah an inherent "interpretive" element to it. The Greeks used the letters to sound exactly as they are written.
I will note that this is all speculation since it is not possible to know what the reasons are for the Greeks changing the direction of writing or why Hebrew specifically started as right-left. Ancient Egypt, for example, had both left-right and right-left and even vertical writing. There are pragmatic reasons for making the switch to left-right writing, for example, most people being right-handed, causing ink to smear less when written from left to right.
The chasam sofer answers this question in his shu"t on OC 187. (You have to read the whole piece in order to understand the linked paragraph, so I'm not going to quote it).
The gist of it though is that there's a gemara in Yuma 59a which discusses how the kohen should sprinkle the blood on the mizbeach. Either he has walk around the mizbeach to the right and go to each of the four corners. Or the opinion holds that the mizbeach was small enough to just stand at one corner and throw the blood to each corner while standing stationary. From this gemara we learn that "every turn one makes should be to the right." Therefore hebrew is written starting at the right side of the page and since we have to end off on the right side (as per the gemara in yuma), the end of a sefer has words written on the right side of the column leaving the left side empty.
According to the other opinion in the gemara where one is stationary, the letters themselves are written from left to right. E.g. a "ר" is written starting from the top left and ends at the bottom right.
(Let me know if my explanation is comprehensible. I'm not sure if I explained it well enough.)