Why does cyclobutadieneiron tricarbonyl behave aromatically?
In the paper "(Cyclobutadiene)iron Tricarbonyls - A Case of Theory before Experiment", Organometallics 2003, 22, 2-20 https://pubs.acs.org/doi/pdf/10.1021/om020946c, at the page 12 nice explanation of bonding in this compound is provided:


  The electronic structure of (cyclobutadiene)iron tricarbonyl has been the subject of many papers. He I and He II low-energy photoelectron spectra provided useful information.77,78 The eight observed bands in the lowenergy PE spectrum of  (cyclobutadiene)iron tricarbonyl in the range 7.65-20.31 eV all were assigned.78 The
  calculations (ab initio SCF MO) showed that there is a net negative charge on the cyclobutadiene ligand that results from π back-bonding from the iron atom into an
  antibonding MO of the ligand (δ bond). Better agreement between the experimental assignments and theoretical calculations was obtained by Chinn and Hall using generalized MO calculations with configuration interaction.79 A simple textbook approach to the bonding is shown in Figure 8.80 It is assumed that it is triplet state cyclobutadiene that is involved with unpaired electrons in the two degenerate ψ2 and ψ3 molecular
  orbitals. These interact with two singly occupied iron orbitals, generating two covalent bonds.