There is no scientific data to support the idea that mixing mineral and synthetic oils will damage your engine (as long as it meets the same American Petroleum Institute (API) and Society of Automotive Engineers (SAE) specifications as regular oil)
When switching from a mineral oil to a synthetic, or vice versa, you will potentially leave a small amount of residual oil in the engine. That's perfectly okay because synthetic oil and mineral-based motor oil are, for the most part, compatible with each other. (The exception is pure synthetics. Polyglycols don't mix with normal mineral oils.)
There is also no problem with switching back and forth between synthetic and mineral based oils. In fact, people who are "in the know" and who operate engines in areas where temperature fluctuations can be especially extreme, switch from mineral oil to synthetic oil for the colder months. They then switch back to mineral oil during the warmer months.
There was a time, years ago, when switching between synthetic oils and mineral oils was not recommended if you had used one product or the other for a long period of time. People experienced problems with seals
leaking and high oil consumption
but changes in additive chemistry and seal material have taken care of those issues. And that's an important caveat. New seal technology is great, but if you're still driving around in a car from the 80's with its original seals
, then this argument becomes a bit of a moot point - your seals are still going to be subject to the old leakage problems no matter what newfangled additives the oil companies are putting in their products.