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The Sun's variability is controlled by the progression and interaction of the magnetized systems that form the 22-year magnetic activity cycle (the "Hale Cycle'') as they march from their origin at $\sim$55 degrees latitude to the equator, over $\sim$19 years. We will discuss the end point of that progression, dubbed "terminator'' events, and our means of diagnosing them. Based on the terminations of Hale Magnetic Cycles, we construct a new solar activity 'clock' which maps all solar magnetic activity onto a single normalized epoch. The Terminators appear at phase $0 * 2π$ on this clock (by definition), then solar polar field reversals commence at $\sim0.2 * 2π$, and the geomagnetically quiet intervals centered around solar minimum, start at $\sim0.6 * 2π$ and end at the terminator, lasting 40% of the normalized cycle length. With this onset of quiescence, dubbed a "pre-terminator,'' the Sun shows a radical reduction in active region complexity and (like the terminator events) is associated with the time when the solar radio flux crosses F10.7=90 sfu -- effectively marking the commencement of solar minimum conditions. In this paper we use the terminator-based clock to illustrate a range of phenomena that further emphasize the strong interaction of the global-scale magnetic systems of the Hale Cycle. arXiv:2010.06048 is a companion article.