(and if headcoverings for men were equally common).

this would never happen, there are two things that push for coverings in Islamic societies, one is religion, other is culture, in Islam men's modesty is relaxed as men are only commanded to from below their navel to above their knee, though there's a difference in opinion regarding that, most scholars believe that navel to knee are parts that must be covered,but in islamic societies men's minimum modesty is their genitals and buttocks, the mainstream opinion is never taken seriously

and of course there comes the culture, and it is easy honestly, the male body isn't sexualized,and male sexual expression is tolerated and isn't associated with the shame/good mechanism, islamic societies don't associate men's sexual expression with honor or dignity, but it associates female sexuality with those values, in fact it associates the honor,manhood and dignity of males by the purity of their females.

Nāmūs is the Arabic word (Greek "νόμος") of a concept of an ethical category, a virtue, in Middle Eastern patriarchal character. Literally translated as "virtue", it is now more popularly used in a strong gender-specific context of relations within a family described in terms of honor, attention, respect/respectability, and modesty.

For a man and his family, namus may, on the one hand, mean sexual integrity of women in the immediate family, their chastity in particular. On the other hand, the man has to provide for his family and to defend the namus of his house, his women in particular, against the threats (physical and verbal) to members of his extended family from the outer world.[2]

Namus of a man is determined by namus of all the women in his family (i.e., mother, wives, sisters, daughters). In some societies, e.g., in Pashtun tribes of Afghanistan, namus goes beyond the basic family and is common for a plarina, a unit of the tribe that has a common ancestral father.[3]

For an unmarried woman, the utmost importance is placed on virginity before marriage. There is a requirement in some cultures for "proof of virginity" (in the form of bloodstains on a bed sheet) to be proudly displayed after the wedding night. Professor of sociology Dilek Cindoğlu writes: "The virginity of the women is not a personal matter, but a social phenomenon".[4]

In the Middle East, for a woman, namus is rooted in obedience, faithfulness, modesty (in behaviour and in dress), "appropriateness".

the double standards exist in both religion and the cultures, so this is very unlikely to happen.