The recently discovered layered kagome metals AV₃Sb₅ (A = K, Rb, Cs) have attracted much attention because of their unique combination of superconductivity, charge density wave (CDW) order, and nontrivial band topology. The CDW order with an in-plane 2×2 reconstruction is found to exhibit exotic properties, such as time-reversal symmetry breaking and rotational symmetry breaking. However, the nature of the CDW, including its dimensionality, structural pattern, and effect on electronic structure, remains elusive despite intense research efforts. Here, we present a comprehensive study on the electronic structure of AV₃Sb₅ by combining polarization- and temperature-dependent angle-resolved photoemission spectroscopy with density-functional theory calculations. Apart from the energy shift of van Hove singularities, we observe double-band splittings for V d-orbital bands in the CDW phase, which provide essential information for revealing the dimensionality and pattern of the CDW order. Our calculations show that three-dimensional CDW orders containing stacking of star-of-David and trihexagonal patterns along the c axis can quantitatively reproduce the experimental features. The characteristic splittings from the two patterns can be experimentally extracted and they are quantitatively consistent with calculations, clearly demonstrating intrinsic coexistence of the two patterns in the CDW order. These results provide crucial insights into the nature and distortion pattern of the CDW order, and its signature in the electronic structure, thereby laying down the basis for a substantiated understanding of the exotic properties in the family of AV₃Sb₅ kagome metals.