The Part 61 FAQ says you can't. But I think the FAQ is wrong. (wouldn't be the first time)
Look at what the regulation says. All of the definitions in 61.1(b)(3) contain one thing in common. They all define a cross country as "time acquired during flight" which meets certain requirements. The "flight" has to meet the requirements, not necessarily the pilot. Of course, the pilot needs legitimate flight time, which is satisfied by being safety pilot.
Let's take an example: the cross country used for most certificates and ratings:
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For the purpose of meeting the aeronautical experience requirements ... for a private pilot certificate ... a commercial pilot certificate, or an instrument rating, ..., time acquired during a flight--
(A) Conducted in an appropriate aircraft;
(B) That includes a point of landing that was at least a straight-line distance of more than 50 nautical miles from the original point of departure; and
(C) That involves the use of dead reckoning, pilotage, electronic navigation aids, radio aids, or other navigation systems to navigate to the landing point.
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Maybe someone else sees in (B) "That includes a point of landing performed by the pilot who logs the time at a point that was at least a straight-line distance of more than 50 nautical miles from the original point of departure." I can't find it.