A color wheel will tell you that red and purple should not be combined. Red and Blue make purple, which means red is purple's mother. Purple and red make magenta, which is a monotone cousin to purple.
The hallmark of interior design is the use of contrasting colors or monotone colors to create an interesting space. That means red and blue or purple and magenta will create a harmonious interior that is pleasing to the eye.
Red and purple are unsettling because they're related, but aren't equal. Red dominates purple, as many mothers tend to do. Of course, there are ways you can liberate purple from its domineering mother. Try using red accents against a purple palette or combine lavender with a bright red for more contrast.
Using purple as an accent against red is more difficult, but can be done by applying the principles of monotone color mixing. Make sure accented purples stand out through texture and select placement rather by color.
Although mixing red and purple presents a design challenge, no one ever said that decorating is only fun when it's easy!