I don't think there's such a thing as thermal transfer inertia, at least at the level of individual atoms (or molecules) within a fluid. In the inertial reference frame of the fluid, each molecule just has its own kinetic energy (and maybe excitation of electrons or whatever) representing its current temperature; I'm not sure what physical state at the molecular level could capture "how rapidly the temperature had recently dropped".
At the macro level, one can come up with ideas. Convection currents within the fluid, and in the air touching the fluid, maybe; though, say, once the hot water has cooled from 100°C to 35°C over a period of—what, ten minutes?—I'm pretty skeptical that there would be so much inertia in those currents that they'd persist, and persist strongly enough to accelerate the 35° to -1° cooling, enough to beat the head start of the water that started at 35°.
Maybe the initially cool water forms some uniform layer of very-cool water at the top (which, being very cool, doesn't exchange much heat with the very-cool air above it; an ice layer would be an example of this), which is held together by surface tension or something; whereas with initially hot water, that layer is not uniform and there's more mixing (and constantly-created convection as a result)? I have no idea if any of that is realistic. If so, it would suggest that shaking a cup of cold water (perhaps after a few minutes in the freezer) would work as well as having it start hot.
... After writing the above, I saw that (a) infogulch below has the same idea, (b) Wikipedia has an anecdote appearing to confirm it[1], and (c) the article doesn't seem to mention shaking, stirring, or otherwise agitating the cup.
[1] The Scottish scientist Joseph Black investigated a special case of this phenomenon comparing previously-boiled with unboiled water;[9] the previously-boiled water froze more quickly. Evaporation was controlled for. He discussed the influence of stirring on the results of the experiment, noting that stirring the unboiled water led to it freezing at the same time as the previously-boiled water, and also noted that stirring the very-cold unboiled water led to immediate freezing.https://en.wikipedia.org/wiki/Mpemba_effect
At the macro level, one can come up with ideas. Convection currents within the fluid, and in the air touching the fluid, maybe; though, say, once the hot water has cooled from 100°C to 35°C over a period of—what, ten minutes?—I'm pretty skeptical that there would be so much inertia in those currents that they'd persist, and persist strongly enough to accelerate the 35° to -1° cooling, enough to beat the head start of the water that started at 35°.
Maybe the initially cool water forms some uniform layer of very-cool water at the top (which, being very cool, doesn't exchange much heat with the very-cool air above it; an ice layer would be an example of this), which is held together by surface tension or something; whereas with initially hot water, that layer is not uniform and there's more mixing (and constantly-created convection as a result)? I have no idea if any of that is realistic. If so, it would suggest that shaking a cup of cold water (perhaps after a few minutes in the freezer) would work as well as having it start hot.
... After writing the above, I saw that (a) infogulch below has the same idea, (b) Wikipedia has an anecdote appearing to confirm it[1], and (c) the article doesn't seem to mention shaking, stirring, or otherwise agitating the cup.
[1] The Scottish scientist Joseph Black investigated a special case of this phenomenon comparing previously-boiled with unboiled water;[9] the previously-boiled water froze more quickly. Evaporation was controlled for. He discussed the influence of stirring on the results of the experiment, noting that stirring the unboiled water led to it freezing at the same time as the previously-boiled water, and also noted that stirring the very-cold unboiled water led to immediate freezing. https://en.wikipedia.org/wiki/Mpemba_effect