Just as balance is key in traditional colour schemes, unorthodox combinations can also work if there is sufficient contrast in tone, value and/or proportion. In this way, two hues which don't fit into any conventional scheme can look great together because they balance each other. Red and aqua is one great example: Red is warm; aqua cool. Red is a deep colour with a medium value, while aqua has a high value.
Some combinations are harder to carry off than others: Red and purple is tricky because both colours have similar values: warm and borderline warm/cool, saturated, deep. Basically they're not different enough, so worn in equal proportions they will fight for attention. But they can work if you make them more different. Take red as the dominant colour and add lilac - a less saturated tint of purple - as an accent, and you have a fresh-looking scheme as the lightness of the lilac lifts the red and helps to balance the overall look. Reverse the proportions and you have a perfect spring outfit given an unexpected pop by the red. Even if we return to fairly vibrant shades of both hues, retaining these proportions can still make the scheme work, so you can accent a purple dress with ruby earrings and red handbag and shoes if you want to, and it looks cohesive enough that no-one will question it - though you may look a bit "When I am old I shall wear purple".
The same principles can be applied to virtually any combination: Rose (a tint of red-violet) and navy (a shade of blue); peach (a tint of orange) and green; violet and spring green - why not?
Another way to deal with tricky colour combinations is to ease in some neutral to help the balance. So while emerald and hot pink is an eye-popping combination, the addition of a muted navy blue, or black, or beige can help moderate these vibrant colours into something a little less extrovert.