It is quite common for a President of one party to select a leading figure of the other party to be part of his Administration.
The number of examples abound, as follows:
John F. Kennedy appointed Republicans including:
Robert McNamara as Secretary of Defense
C. Douglas Dillon as Secretary of the Treasury
Henry Cabot Lodge Jr as Ambassador to South Vietnam
Lyndon B. Johnson appointed Republicans including:
John W. Gardner as Secretary of Health, Education and Welfare
Henry Cabot Lodge Jr as Ambassador to West Germany
Richard Nixon appointed Democrats including:
Sargent Shriver as Ambassador to France
John Connally as Secretary of the Treasury
Gerald Ford appointed Democrats including:
Daniel Patrick Moynihan as Ambassador to the United Nations
Jimmy Carter appointed Republicans including:
James Schlesinger as Secretary of Energy
William H. Webster as Director of the FBI
Ronald Reagan appointed Democrats including:
Mike Mansfield as Ambassador to Japan
Jeane Kirkpatrick as Ambassador to the United Nations
William Bennett as Secretary of Education
George H W Bush appointed Democrats including:
Robert Strauss as Ambassador to the Soviet Union/Russia
Bill Clinton appointed Republicans including:
William Cohen as Secretary of Defense
George W. Bush appointed Democrats including:
Norman Mineta as Secretary of Transportation
Barack Obama appointed Republicans including:
Robert Gates as Secretary of Defense
Jon Huntsman Jr. as Ambassador to China
Robert Mueller as Director of the FBI
Ray Lahood as Secretary of Transportation
Chuck Hagel as Secretary of Defense
Donald Trump appointed Democrats including:
Gary Cohn as Director of the National Economic Council
Joe Biden appointed Republicans, including:
Christopher Wray as Director of the FBI
Jerome Powell as Chairman of the Federal Reserve
Jeff Flake as Ambassador to Turkey
Meg Whitman as Ambassador to Kenya