By halves I am going to assume this is referring to the administrative split of the Roman Empire into East and West performed by the Emperor Theodosius I in 395 AD (at his death).

While the two halves of the Empire after 395 AD increasingly went separate ways politically, they still largely thought of themselves as part of a greater whole. There was certainly rivalry between Emperors and generals from the two halves but nothing that came to full scale conflict. In fact, the Eastern Roman Empire (ERE) provided military assistance to the Western Roman Empire (WRE) as late as 467/468 AD during the disastrous invasion of Vandalic Carthage/Libya. The WRE would officially cease to exist in 476 AD.

The closest the WRE and ERE came to blows was during a rather complicated time period where Flavius Stilicho, the highest ranking general in the west, attempted to use the Visigoths and their king Alaric to threaten the ERE over territory in Illyricum. Stilicho didn't live long enough to follow through on this and his end goals/plans for Alaric and the Visigoths are not entirely clear.

Edit thanks /u/Guckfuchs for correcting me, I had completely forgot about the succession crisis after Honorius' death. The Eastern Emperor Theodosius II did send an army to remove Johannes (John) from the Western Throne and installed Valentinian III, his cousin. It is an important and interesting point to note that Theodosius bothered naming a new Emperor in the West, rather than simply claiming sole control over the entire Roman Empire.

History of the Wars Procopius

Historia Nova Zosimus

I’d like to add a small correction to u/mrleopards ‘ and u/WispyWalrus ‘ accounts. There was at least one instance of armed conflict between the two halves of the empire after 395 AD. In August 423 AD the western emperor Honorius died. Throughout his reign there had been several moments when both halves of the empire nearly came to blows even though the east had been ruled first by his brother Arcadius and then his nephew Theodosius II. Honorius himself had never much control over governance. Instead the west was ruled by powerful military figures like Stilicho and later Flavius Constantius. The later even rose to co-emperor in the west. Both men seem to have been close to starting a war with the east at times but both also died before anything could come about. And it had always been Honorius who had guaranteed the legitimacy of the western government as he was a member of the dynasty that had been ruling both parts of the Roman Empire for more than half a century.

So when he finally died in 423 AD that posed something of a problem for the western elites: they didn’t have any members of the Valentinian-Theodosian family left which could have ascended to the throne. After a few months of negotiations they agreed to elevate a senior civil servant, the primicerius notariorum Johannes, who became emperor in November of the same year. He had the support of powerful generals like the cura palatii Aetius. Others however were much colder towards him and so his control over the west was sketchy at best. The most important opposition towards Johannes came from the east. The court of emperor Theodosius II wasn’t thrilled to see the western throne pass to someone who wasn’t part of the dynasty. Johannes and his supporters certainly hoped to come to an agreement with Constantinople but when Theodosius elevated his cousin Valentinian III to the purple in October 424 AD it became clear that civil war was inevitable.

Valentinian who was just a five year old boy at the time was the son of Galla Placidia, the half-sister of Honorius and Arcadius. She had been an influential figure at the western court until court intrigues had forced her to relocate to Constantinople in 422. Now she and her son returned to the west at the head of an East Roman army commanded by the general Ardaburius. After some initial successes disaster struck in the form of a storm that hid the East Roman navy. Ardaburius himself was captured and taken prisoner at Ravenna. However his son Aspar who had been in command of the cavalry and was leading them into Italy by land could again change the fortunes of war. He was able to take Ravenna, free his father and deliver Johannes to Galla Placidia and Valentinian III at Aquileia. When Aetius arrived in Italy with a substantial contingent of Hunnic soldiers his emperor had already been executed and the civil war was effectively over. Aetius was able to come to an agreement with the new regime and would later rise to become the dominant military figure in the west. The hero of the war, Aspar, achieved a similar ascent in the east.

There are some famous late antique monuments that are connected to this rather short civil war. At Ravenna Galla Placidia founded the church of San Giovanni Evangelista. The mosaic decoration of its apse is unfortunately lost but according to an early medieval source it commemorated Placidia’s miraculous rescue from the storm that had hit the eastern army. It also featured several portraits of members of her family, the dynasty who’s rule over the west had been restored in 425. In the east it is possible that the Golden Gate, the huge marble main entry into Constantinople, was also constructed to commemorate victory in this civil war. It was originally topped by a bronze statue of Theodosius riding a chariot drawn by four elephants. Although the emperor had never left his capital throughout the events he was seemingly still able to propagate himself as the true victor of the war.

Finally, to echo u/shlin28 here: It is important to remember that those events as well as similar wars that preceded them shouldn't be seen as conflicts between two distinct states but as civil wars over the leadership in one unified empire. Theodosius and his generals weren’t fighting an enemy state but a usurper who had deprived his family of its right to rule both halves of the empire.