WASHINGTON, Aug 1 (Reuters) - Former U.S. Defense Secretary Donald Rumsfeld said on Wednesday the family of Army Ranger and football star Pat Tillman deserved the truth about his death by friendly fire in Afghanistan but did not initially get it.

Rumsfeld's testimony, his first appearance on Capitol Hill since leaving office last year, came a day after the U.S. Army censured retired Lt. Gen. Philip Kensinger, former head of the Army Special Operations Command, for lying about Tillman's death by American fire in Afghanistan in 2004.

"The handling of the circumstances surrounding Corporal Tillman's death could only have added to the pain of losing a loved one," Rumsfeld told a congressional hearing.

Tillman became a patriotic symbol in the United States when he gave up a multimillion National Football League career to join the Army after the Sept. 11, 2001, attacks on New York and Washington.

He died on April 22, 2004, in Afghanistan from what the Army initially said was enemy fire and was later awarded the Silver Star with a citation that suggested that he was killed by enemy fire.

Kensinger, who may be demoted in the case, refused to testify at Wednesday's hearing. He rejected the Pentagon's allegations in a statement on Tuesday, saying: "Never did I lie or would I lie, deceive or intend to obstruct or mislead in any fashion."

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