Margaret of Austria and Mary of Austria ruled there successively between 1507 and 1555, and while female rulers weren't unknown in Europe certainly it is fair to say they were uncommon and my impression is that it certainly could color perspectives on the appropriateness of their position and support for it.
In both these cases though, they were appointed, which certainly speaks to the Emperors themselves feeling it was the correct choice of Governor-General. What was the decision process behind it?
And as in the title, once placed in power, how did their gender impact their rule, both in terms of how others viewed their legitimacy, but also how they themselves worked to counter any opposition that might have been based on such objection?
When Maximilian of Habsburg, Margaret of Austria’s father, appointed her to be governess, she had been engaged once and married twice. When her last husband, Philibert of Savoy died in 1504, she had had enough and vowed to remain a widow until the end of her days. She refused to remarry, the plan that was put forward by her father first. Then, her brother, Philip the Fair and the lord of most of the principalities that made up the Low Countries, died in 1506. Since Margaret did not want another marriage and had no children, Maximilian had to find another role for her, and at her insistance, he appointed her governess of the Low Countries.
The main reason for her appointment was not that she was extremely qualified for the job, although she was. Her mentors had been Louise of Savoy, queen-regent of France and Margaret of York, duchess-dowager of Burgundy, two of the most famous female political minds of her time. Maximilian just thought he could control her, and use her as a figurehead only, while she took care of the childeren of her brother, including the heir apparent, the future Emperor Charles V.
It was most of all her widowhood that made Margaret accepted as regent for Maximilian and later Charles (he became king of Castile in 1518 and reluctantly reappointed his aunt as governess). Noblewomen were expected to serve the family by having and taking care of children. Since Margaret had no children of her own, another role had to be found for her, and that happened to be to rule as regent in the Low Countries until the true heir, Charles was old enough to rule by himself.
Maximilian could not take up this role himself since he was busy in the Holy Roman Empire securing the imperial throne although he remained regent in name. In addition, Margaret was a direct descendent of the ruling family in the Low Countries, being the daughter of Mary of Burgundy, the last of the Valois duchesses of Burgundy. Maximilian on the other hand, was percieved as a foreigner, only having married into the Valois family. His constant meddling in the affairs of the Low Countries, however, was a great point of contention between him and his daughter.
Mary of Hungary similarly was a widow when she was appointed governess in 1531 by her brother Charles V. She too had had an excellent political education, this time by her grandfather Maximilian himself. The fact that she had served as queen of Hungary until her husband died in the battle of Mohacs in 1526, had given her some ruling experience as well. She, more than Margaret was percieved as a capable political and even military mind. Her acceptance as governess again had to do with her being a descendant of the dynasty, being the sister of the Emperor himself. She too was a widow, and as such suited for the role as governess (in the correspodence about her appointment this is the arugment that convinced Charles the most).
More than Margaret, who mostly relied on her status as a widow, Mary played up her manly attributes, by going hunting for instance. This to make her associated with the image of the virago, or man-woman, in essence a male spirit in a female body. Since they could not truly gain prestige from manly persuits, however, both Margaret and Mary invested in culture, making their courts renowned for art and book collections. They also stimulated the arts by appointing famous artists to their courts. Finally, they corresponded with the most influential humanists of their time, all to enhance their credibility as leaders.
Finally, it is important to remember that the governesses were never truly independent from their male counterparts. They derived their authority from the head of their family, and in principle could not act without his approval. Charles was livid for instance when Margaret decided to negotiate an independant treaty between the Low Countries and France in the 1520ies, something that soured the relationship between him and his aunt until her death in 1530. Mary seemed to have been more complacant, but Charles had made sure that she could not act as independent as her predecessor by reforming the government system in the Low Countries.
I hope this extremely rambling and desperatly superficial response is an answer to your question. If you would like to read more about this subject, i recommend authors like Dagmar Echberger, Jacqueline Kerkhoff (if you are fluent in Dutch or French) and Lorraine Attreed (in English but she mostly wrote about the cultural aspects of the court of Margaret of Austria). I would also be happy to elaborate more on some of the things I have written!