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There are a number of common sentence patterns for inviting someone to do something.
1
odorimasenka. "Will you dance?" Literally, it means "Won't you dance?". Oddly, the two English translations, although apparent opposites, mean the same thing.
The verb suffix ...masenka is a polite invitation to do something; the implication is that the speaker is already doing (or already plans to do) something and is inviting the listener to participate. The ruffians are already dancing, and invite Sabrina to join them.
watashitotomodachihaashitaatorantisuniikimasu.ikimasenka. "We're going to Atlantis tomorrow. Would you care to come too?" (The ruffians invite Sabrina to participate in their plans.)
Confusingly, it could also be a plain question about the listener's plans not to do something. odorimasenka could equally validly be translated as "Are you not going to dance?", with no implication that the speaker will participate too.
warumonohaashitaatorantisuniikimasu.ikimasenka. "The ruffians are going to Atlantis tomorrow. Are you not going too?" (The wizard notices that Sabrina is not packing her bags and wonders why.)
Even more confusingly, it can also be a suggested action. odorimasenka could equally validly be translated as "Won't you dance?", this time with the implication that the speaker would enjoy watching the listener dance, not that the speaker wants to participate too.
warumonohaashitaatorantisuniikimasu.ikimasenka. "The ruffians are going to Atlantis tomorrow. Why don't you go too?" (The head ruffian thinks that his gang may need assistance in Atlantis tomorrow and is politely instructing Sabrina to give it.)
2
odorimashyou. "Let's dance!" This pattern is used when the speaker is responding to an invitation (given using the ...masenka pattern), thinks that he already knows that the listener wants to dance, or does not really care what the listener wants to do.
ashitaatorantisuniikimashyou. "Let's go to Atlantis tomorrow." The wizard has zapped Sabrina with a charm spell which makes her amenable to suggestions, and knows that she must agree.
3
odorimashyouka. "Shall we dance?" The implication is that the speaker is inviting the listener to do something together. It is not used when inviting the listener to join in a current (or currently-planned) activity.
ashitaatorantisuniikimashyouka. "Shall we go to Atlantis tomorrow?" The head ruffian has no plans for tomorrow, and invites Sabrina to accompany him to Atlantis. Since she is in his gang, he expects her to agree.
Confusingly, this pattern also has a second interpretation.
odorimashyouka. "Shall I dance?" The speaker is proposing to dance and is asking for permission or approval.
warumonohaashitaatorantisuniikimasu.ikimashyouka. "The ruffians are going to Atlantis tomorrow. Shall I go too?" Sabrina, bored by sitting in the gang headquarters, is offering to accompany the ruffians on their adventure.
4
odo,taradoudesuka. "How about dancing?"
This pattern suggests an action and asks for opinions, without as yet issuing an invitation to anyone to do anything.
ashitaatorantisunii,tadoudesuka. "How about going to Atlantis tomorrow?" Sabrina, bored by sitting in the gang headquarters, wants to do something exciting.
5
Note also that some of the patterns for expressing preferences and recommendations can also sometimes be interpreted as offers or invitations.