Descartes' rule of signs, canonical sign patterns and rigid orders of moduli

Abstract: We consider real polynomials in one variable without vanishing coefficients and with all roots real and of distinct moduli. We show that the signs of the coefficients define the order of the moduli of the roots on the real positive half-line exactly when no four consecutive signs of coefficients equal $(+,+,-,-)$, $(-,-,+,+)$, $(+,-,-,+)$ or $(-,+,+,-)$.