TAIPEI—When Taiwan’s new president takes the oath of office this week, two vital constituencies—one at home, the other across the strait in mainland China—will be listening hard, but for different things.

Tsai Ing-wen heads a political party nominally in favor of Taiwan’s independence from China, a red line for Beijing that would formalize a split from a stalemated civil war more than six decades ago. At Friday’s inauguration, she is expected to reiterate her campaign pledge to maintain the status quo, even though many voters...