We found that the conventional model of orbital ordering of 3x 2 − r 2 /3y 2 − r 2 type in the eg states of La0.5Sr1.5MnO4 is incompatible with measurements of linear dichroism in the Mn 2p-edge x-ray absorption, whereas these eg states exhibit predominantly cross-type orbital ordering of x 2 − z 2 /y 2 − z 2 . LDA+U band-structure calculations reveal that such a cross-type orbital ordering results from a combined effect of antiferromagnetic structure, Jahn-Teller distortion, and on-site Coulomb interactions.PACS numbers: 75.47. Lx, 75.30.Mb, 71.28.+d, 78.70.Dm Orbital ordering, which manifests itself in the spatial distribution of the outermost valence electrons, is an important topic in current research of transition-metal oxides, as the magnetic and transport properties are closely related to the orbital and charge degrees of freedom [1]. In particular, charge-orbital ordering of half-doped manganites has attracted much attention [2,3,4,5,6,7,8]. The mechanism of charge-orbital ordering is being hotly debated [9,10,11,12,13,14,15]. To observe orbital ordering directly is a difficult task. Experimental results of resonant x-ray scattering (RXS) at the Mn K-edge in La 0.5 Sr 1.5 MnO 4 and LaMnO 3 show removal of degeneracy between 4p x and 4p y ; these observations have been argued to be direct evidence of orbital ordering [16,17]. However, the origin of RXS at Mn K-edge is controversial. Orbital ordering in transition-metal oxides is typically accompanied by Jahn-Teller lattice distortion. Calculations based on a local-density approximation including on-site Coulomb interactions (LDA+U) [18,19] and multiple scattering theory [20] indicate that RXS measurements pertain mainly to Jahn-Teller distortion, instead of directly observing 3d orbital ordering. Multiplet calculations have shown that one can use linear dichroism (LD) in soft x-ray absorption spectroscopy (XAS) to identify the orbital character of 3d states in orbitalordered manganites [21].Half-doped manganites such as La 0.5 Sr 1.5 MnO 4 exhibit CE-type antiferromagnetic (AFM) ordering and charge-orbital ordering [16,22,23]. They have a zigzag magnetic structure in which the magnetic moments of Mn on the chain form a ferromagnetic coupling, but AFM coupling between the zigzag chains. Below the charge-ordering (CO) temperature T CO =217 K, the valence of La 0.5 Sr 1.5 MnO 4 orders in an alternating pattern with two distinct sites identified as Mn 3+ and Mn 4+ [22,23]. The e g electrons on the nominal Mn 3+ sites of La 0.5 Sr 1.5 MnO 4 are believed to exhibit an orbital ordering of 3x 2 − r 2 /3y 2 − r 2 , in which occupied d 3x 2 −r 2 and d 3y 2 −r 2 orbitals are alternately arranged at two sublattices in the ab plane [7]. However, d 3x 2 −r 2 and d x 2 −z 2 (d 3y 2 −r 2 and d y 2 −z 2 ) orbitals might be mixed, because orbitals of these two types have the same spatial symmetry with respect to the MnO 2 plane. To clarify the nature of orbital ordering in La 0.5 Sr 1.5 MnO 4 is essential to reveal the origin of orbital ordering in half-doped manganites...