Eunuchs have played major role in the history of China, at times making up a significant part of the bureaucracy. Their political value, to me, seems apparent; if a position of power is inhabited by someone who is unable to have children, then that position can be filled at the discretion of the ruler by the most qualified candidate, not the candidate of the highest birth. Yet I haven't heard much about eunuchs being present at all in the Roman Empire. Have I just not done enough reading? Were they a significant force?
By the way, sorry about the tasteless phrasing of the question. I couldn't think of a better way to put it.
The difference between China and Rome was that whilst the Chinese examination system nominally produced a class divide by way of meritocracy (Balazs 1964), Imperial Roman society had clear class divides – slaves, freedmen, ordinary plebeians, equestrians, and senators. All of these bodies could be quite sizeable – there was an ideal number of 600 senators, for example, hence the revision of senatorial lists, e.g. by Augustus (Augustus, Res Gestae Divi Augusti 8) Domitian (Suetonius, Domitian 8), and so as emperor one did have quite a bit of luxury as to whom to give responsibilities to. One need only look at the case of Pliny the Younger, a senator assigned as governor of Bithynia by Trajan (Pliny, Epistles), or Frontinus, another senator who, having served a brief stint as governor of Britain, was made water commissioner by Nerva (Frontinus, On the Aqueducts).
Additionally, posts were not hereditary to begin with, as the Principate inherited the framework of the late Republic. Hence, governorships, civil offices and so forth were never held for long. senators might not have senator children (nor, as noted, stay senators for long themselves), and it was possible for people of 'new families' to enter the Senate – Frontinus above is a case in point. However, a general lack of social mobility could prove highly useful for emperors wary of senators' ambitions.A key point of Augustus' imperial policy was the assignment of equestrians rather than senators to the governorship of Egypt, as they would (presumably) be less willing to try and use Egypt as a base of operations for a takeover of Rome. Later, Claudius made heavy use of freedmen in the day-to-day running of the empire, as they could at least be expected to remain loyal (Tacitus, Annals Book XII).
In any case, unlike China, the retention of staff in the wake of a succession was rarely certain. Claudius' freedmen disappeared from the record when Nero came to power, whilst Macro, the Praetorian prefect who allegedly killed Tiberius on Caligula's orders, was himself arrested and committed suicide soon after.
It must also be noted that there was social stigma surrounding eunuchs and male castration, and it was indeed illegal for a Roman citizen to castrate himelf. Augustan poetry on the Battle of Actium at times derides Cleopatra's eunuch thralls, for example, and the senate had to decree that priests of the cult of Cybele (a.k.a. Magna Mater – the Great Mother), the Galli, could not be male citizens, as it was required that they castrated themselves. Domitian banned the practice of male castration entirely (Suetonius Domitian 7) and had his cousin Domitilla, a probable Cybele worshipper, expelled as part of a general effort against foreign religion (Suet. Dom. 8).