She is a proxy in debates over feminism and political power and a recurring subject of parody on “Saturday Night Live.” Views about her — particularly on issues of trust after the F.B.I. investigation into her handling of emails as secretary of state — are seared in place with many voters, pollsters say. After 25 years, her days of reinventing herself are almost surely gone.

And yet these four nights in the public eye, particularly coming after Mr. Trump’s often chaotic convention in Cleveland, offer her a chance to persuade important segments of the electorate to take another look at her, to consider parts of her life that have been lost in the glare of scandals and investigation, and to judge her candidacy in contrast with the dark image Mr. Trump presented last week of himself and the nation.

That would start with supporters of Senator Bernie Sanders of Vermont, many of whom, particularly younger ones, learned much of what they know about Mrs. Clinton through the prism of Mr. Sanders’s attacks and have been steadfast in their opposition to her — potentially all the more so after her choice of Senator Tim Kaine of Virginia, a moderate, as her running mate.

But just as crucially, Mrs. Clinton might also find an audience with independent voters struggling between two candidates they do not like, who will have an opportunity after this week to make a direct comparison between the visions being offered by Mrs. Clinton and Mr. Trump.