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I have just finished my empirical test of your hypothesis, Karl. I can assure you that you are wrong.
After unravelling some 300 metres of my wife's gardening twine I find that the little ball of stuff did indeed have two ends. To test my results I snipped a metre off one end and, voila!, I produced two strings, both of which had two ends. My problem at this point in time is to put this mini-universe back together again (yes, I've nicknamed the problem 'Humpty').
Any and all practical advice you might be able to give me to attain this all consuming goal would be well received. Time constraints apply as the good wife will be home in 8 hours or so.
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