Word Reference's Italian dictionary gives the following pronunciations:
Is this correct? If so, is it usual that the pronunciation of a vowel of an adjective is different in the corresponding adverb with the -mente suffix?
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I assume you are talking about the pronunciation of the E in the second syllable. The change from [ɛ] in [sinˈtʃɛro] to [e] in [sintʃeraˈmente] is completely regular and predictable, because "standard"* Italian pronunciation includes a rule saying that [ɛ] and [ɔ] are only possible in the stressed syllable of a word. When suffixation causes the stress to shift to another syllable, the formerly stressed [ɛ] and [ɔ] turn into [e] and [o] respectively.
If the stressed syllable contains a rising diphthong such as [jɛ] or [wɔ], sometimes the entire diphthong is replaced by [e] or [o] when the syllable becomes unstressed. But other times the glide remains, as in the diminutive cuoricino [kworiˈʧino] from cuore [ˈkwɔre]. (The form coricino is apparently archaic or dialectal.) Unlike the change of [ɛ] and [ɔ] into [e] and [o], the loss of the glide element of the rising diphthongs [jɛ] and [wɔ] in unstressed syllables is not an automatic or exceptionless process in modern Italian. (I recently read a paper that tried to find patterns to it, but I don't remember right now exactly what it said.) I can't think of a -mente word where a diphthong is converted into a single vowel; I only mentioned this type of alternation because it is somewhat related in a more general way to the topic of your question.
*Note: The concept of a "standard" for pronunciation is rather tricky. Unlike spelling, which is about the use of discrete letters, pronunciation involves sounds, which can vary on a continuous basis. I'm just using "standard" to refer to the rules of pronunciation that are commonly described as standard, or taught to learners.