The Holy Spirit in one word – Fellowship
The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all.
2 Corinthians 13:14 ESV
The one word used here for the Holy Spirit is fellowship, which can also be translated as communion. So what it the fellowship or communion of the Holy Spirit? That really does not depends on your church tradition, but we have to be aware that different traditions use the words in different ways.
The Reformed Tradition:
When you join or are baptised into many churches in the Reformed tradition you are offered the Right hand of fellowship. This can be just another bit of Christian jargon, which as it is not used in other traditions in this way is a way of confusing things. But it does have meaning. It means that you have a commitment to the congregation and they have a commitment to you and together you are committed to Jesus Christ. The right hand of fellowship is cone with great feeling and sincerity, but that can be lost to those who do not understand this jargon.
The Catholic Tradition:
More jargon, this time my usual bit about the Greek words translated. Fellowship is the English translation of koinonia, which is a togetherness that goes beyond individuality. It was translated from Greek to the Latin communion, and from there to the English communion. It is from this we get the communion of the saints, the idea that all Christians, those alive and those who have died, are one in communion or fellowship with God. Some avoid the word communion here as it can be confused with Holy Communion.
Putting it together
As I am an Anglican, a tradition which is Catholic and Reformed, I can have my cake and eat it here. I could believe both meanings above held in flux. But to be honest, I can’t, I find the Reformed meaning to be a little troubling as it is too narrow in its scope.
The Fellowship of the Holy Spirit is not as narrow as to refer to a single denomination or the even narrower single assembly. There is a commitment to the congregation and the congregation to us, and of both together to God, but here the congregation is the great communion of all believers. All Christians who are living, have lived or will live.
So there is no excuse not to accept the other church up the road who do things differently to us. They might catch people for Christ that we cannot. Especially if we do not bicker.