Is it important to have a centered stereo image?

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So I was just playing around with the Inspector plugin for the first time and I'm very disturbed by what the stereo meter says about some tracks I've been working on. Three separate tracks all pull to the right although they sound perfectly centered to me. Am I going deaf in one ear?

I then listened to a wide variety of other tracks and noticed that many more modern tracks are pretty straight down the middle on the meter but some of the better sounding older songs tend to wander back and forth from left to right. I think I only found one other song that pulled to the side like mine.

So I'm wondering, how much does this matter? And how does this stereo meter actually work? How does it account for frequency differences? Is it possible that I'm putting some low-end on one side and compensating with a high-end sound on the other so that it sounds balanced even if the energy on each side is different? I can't believe it's only my hearing because I can listen to a tone straight down the middle and it sounds perfectly centered. But I suppose I should get my hearing checked