An asymmetric H-shaped model polybutadiene (PBd) melt, H(SS13LL20)B58, with a backbone of molar mass 58 kg/mol and with two short arms of molar mass 13 kg/mol and two long arms with molar masses around 20 kg/mol, was designed and synthesized by anionic polymerization and purified by fractional precipitation. To obtain this novel structure, two short arms were synthesized and linked together using 4-(dichloromethylsilyl)diphenylethylene (DCMSDPE). The backbone was then grown from this linkage point to produce an asymmetric star intermediate terminating in a reactive anion at the end of the backbone. The two long arms were similarly grown in a separate reaction vessel and then linked by DCMSDPE, and these linked long arms were then attached to the backbone of the asymmetric star. The multiple steps of this reaction scheme led to multiple possible byproducts. The most likely of these were identified and quantified by temperature gradient interaction chromatography (TGIC) performed on both purified and unpurified H(SS13LL20)1358 product and its asymmetric star-shaped synthetic precursor. We then performed linear rheological studies on purified and unpurified H(SS13LL20)B58, its star precursor, and blends of the purified H polymer with its star precursor and found that their rheological behaviors can be predicted reasonably accurately by the "hierarchical model" [Wang et al. (2010)], if the impurities identified by TGIC are accounted for in the rheological modeling. These results show that TGIC characterization of product and intermediate reaction products, supplemented by rheological studies of controlled blends, are important steps in determining the accuracy of rheological models of polymers with complex branching architectures.