If $X$ is a spectrally positive Levy process, $\bar{X}^c$ the continuous part of its maximum process, and $J$ the sum of the jumps of $X$ across its previous maximum, then $X - 2\bar{X}^c - J$ has the same law as $X$ conditioned to stay negative. This extends a result due to Pitman, who links the real Brownian motion and the three-dimensional Bessel process. Several other relations between the Brownian motion and the Bessel process are extended in this setting.