I have found the solution. Thanks to your comment, esmi, and a little testing on my part, it can simply be done with CSS and the WordPress tags. It will display just as a normal Next or Previous link, no extra special CSS skills required. ;)
All I probably need to say (and WordPress could do with adding on their Codex) is you can add HTML inside the next_post() and previous_post() tags. CSS is just applied to it as you would a normal HTML element. I built my arrows in another page, made them look right and copied the code into their appropriate place. Since I used single quotes ( ' ) to define the variables in the PHP, double quotes ( " ) can be used to define the HTML attribute values as normal, and the % works just to add the actual link, as normal. No idea how to get the link to wrap around the entire HTML code, but I will leave it at that. It works this way, so I am happy.
Thanks for your help. I don't think I would have tried had you not said that it was just HTML coding.