I wrote an answer to an earlier, similar question here, focusing on the Spanish Jesuit José de Acosta's description of Latin America's warmer climate which were published in the late 16th century. In the thread to I linked above u/khosikulu interestingly discusses Pigafetta's account of Magellan's circumnavigation, which is maybe a bit closer to describing a somewhat similar description of the reversed seasons.

I'll add my summary from there to add context on how Acosta's views differed from the classical European canon:

The example of José de Acosta has shown first that the new experiences of the Americas led him to differ from Aristotle's highly influential views on meteorology. He described weather phenomena in South America that would have been impossible according to Aristotle: Including the hospitabilty of the lands despite being so far south, and his feeling cold despite the sun shining overhead. He did not make a reversal of the seasons more explicit in his work. These new views however did not mean that all classical knowledge was discarded with. The canon of bible and classical authors (incl. the geocentric model) remained an important frame of reference during this time period. Nonetheless, authors like Acosta out of necessity (due to the "newness" of the "discoveries") initiated a trend of turning to their own voice rather than traditional authorities to confirm the authority of their writings for European audiences who were far from the Americas.

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[Edit:] Going back now to Acosta's text, I found that he actually did mention quite clearly the reversal of the seasons, and attempted to give explanations for them that fit within the intellectual climate of his times. I'm editing my post to add this perspective below.

​ Among the possible explanations José Acosta gives for the reversal of the seasons are: The strong rains ( connected to the sun), the shortness of the day (to explain the surprinslgly moderate temperatures in many parts), as well as other meterological explanations focusing on the ocean and winds that would cool the temperature. Overall we see here a rather "proto-scientific" mentality at work, focusing on experiences and concrete examples, as well as comparing with similar phenomena in Europe and Africa. It's important to note again though, that as a Jesuit and someone living in the 16th c., it's not surprising that Acosta intersperses his accounts with mentions of how this climate shows the workings of God or a divine spirit. And while he directly challenges many ideas of classical philosophers incl. Aristotle, he also ackowledges that they are still right on other points.

I'll now give some quotes from Acosta's *Historia natural' giving the most important of his explanations, starting with the influence of the rains:

Considering with my selfe often times what should cause the Equinoctial to be so moist, as I have said, to refute the opinion of the Ancients, I finde no other reason but the great force of the sunne in those partes, whereby it drawes unto it a great aboundance of vapors from out of the Ocean, which in those parts is very great and spatious; and having drawne unto it this great aboundance of vapours, doth suddenly dissolve them into raine, and it approoved by many tryed experiences, that the raine and great stormes from heaven proceed from the violent heat of the Sunne; [....] but when the Sun retyres, the heat is moderate, and then there falls no raine; whereby we may conclude that the force and heat of the Sunne is the cause of raine in those countries. [...] [T]he great draught which the Ancients held to be in the middle region, which they call the burning Zone, is nothing at all; but, contrariwise, there is great humiditie, and then it raines most when the sunne is nearest.

In this chapter he notes the moderate temperatures in most parts of South America, despite them lying under the equinox. Here they are connected again with rain but also the shortness of the days:

This presupposed, if any one demaund of me, why under the Equinoctiall Line the heat is not so violent in summer as in some other Regions (as in Andalusia in the moneths of Iuly and August), I will aswere, that in Andalusia the dayes are longer and the nights shorter; and as the day being hot, inflames and causeth heat, so the nights being cold and moist, give a refreshing. [...] And although the burning Zone be nearer the Sunne then all other Regions, yet doth not the heate continue there so long. [...] He therefore that shal put these two properties of the Zone in one ballance, that it is most rainie in the time of greatest heate, and that the dayes are shortest there, he shall perchance finde them to equall the other two contrarieties, which bee, that the Sunne is neerer and more directly over them then in other Regions.

Further reasons for differences in temperatur between parts of "Old" and New World" that lie on a similiar altitude include for Acosta:

The first is the Ocean, the second the situation of the land, and the third, the nature and propertie of many and sundry windes. [...] I have first placed the Sea, for without doubt, the neerenesse thereof doth helpe to temper and coole the heat; and commonly Countries lying neere the sea are cooler then those that are farther off. Caeteris paribus, as I have said, even so the greatest part of the new world, lying very neere the Ocean, wee may with reason say, although it bee under the burning Zone, yet doth it receive a great benefite from the sea to temper the heat.

He then goes into further discussions of the incluence of cold winds and the nature of the land to explain the temperate climate in parts of South America. As noted, there's an overall clear tendency of focusing on experience to explain such wether and climate phenomena, all the while praising God for these workings. Acosta was a member of the Jesuits, a religious order that would continue to focus strongly on scientific methods, and also use European scientific developments to aid in conversion campaigns, in order to showcase the Christian god's supremacy (e.g. in China or Korea). Last but not least, I should note that Acosta's Historia Natural (published in the late 16th c.) circulated widely and proved to be a major source for other authors on the Americas throughout the 17th century at least.

I'll close with the last title of part two of his book, which sums up many of his arguments quite nicely:

Ch. XIV. - That they which inhabite under the Equinoctiall, live a sweete and pleasant life.

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• I used a translation of Acosta's Historia natural from 18801604, which explains the Olde English (transl. by Grimeston, Edward; ed. in 1880 by Markham, Clements Robert) [edit: thanks to u/swuboo for clearing up the publication date]. It's accessible online via archive.org, the quotes come from libro II, chapters VIII ff.

Edit: Corrected an earlier misinterpretation of Acosta, and added quotes from him on the reversal of seasons

Edit 2, 3: Added the source, edited its publ. date

Follow up. Would they be able to figure it out considering the travel times involved in traveling between the hemispheres back then?