The coloured chemical in carrots is beta carotene which is not very water soluble. In other words it's kind of oily. Oil and water do not mix. The red colour of beets is from anthocyanin which is soluble in water. If you put sliced carrots and beets in some clear vegetable oil you might get the opposite effect--although the anthocyanin might also be somewhat solouble in oil so it might not work perfectly. For sure you will get a yellow/orange tinge to the oil from the beta carotene in carrots if you have small pieces and you leave it in long enough, maybe heat it a bit.
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