Isn't it the 10Y-2Y which is the famous recession signal?
[0] https://www.frbsf.org/economic-research/publications/economi...
The synthetic instrument 3m-10y is referred to as a 'curve'. It can be denoted 3m-10y. Convention is such that the value of a curve is long dated tenor minus short dated tenor.
One names the spread by denoting the two instruments, by convention first the short dated, then the long dated (for example, 2y/10y or 3m/10y or indeed 3m-10y, where the '-' is a hyphen).
One computes the spread by subtracting the short dated yield from the long dated one (ie, 10y - 3m, where the '-' is a minus).
It contains gems such as this:
> Section 4.11. FRN Convention; Eurodollar Convention. “FRN Convention” or “Eurodollar Convention” means, in respect of either Payment Dates or Period End Dates for a Swap Transaction and a party, that the Payment Dates or Period End Dates of that party will be each day during the Term of the Swap Transaction that numerically corresponds to the preceding applicable Payment Date or Period End Date, as the case may be, of that party in the calendar month that is the specified number of months after the month in which the preceding applicable Payment Date or Period End Date occurred (or, in the case of the first applicable Payment Date or the Period End Date, the day that numerically corresponds to the Effective Date in the calendar month that is the specified number of months after the month in which the Effective Date occurred), except that (a) if there is not any such numerically corresponding day in the calendar month in which a Payment Date or Period End Date, as the case may be, of that party should occur, then the Payment Date or Period End Date will be the last day that is a Business Day in that month, (b) if a Payment Date or Period End Date, as the case may be, of the party would otherwise fall on a day that is not a Business Day, then the Payment Date or Period End Date will be the first following day that is a Business Day unless that day falls in the next calendar month, in which case the Payment Date or Period End Date will be the first preceding day that is a Business Day and (c) if the preceding applicable Payment Date or Period End Date, as the case may be, of that party occurred on the last day in a calendar month that was a Business Day, then all subsequent applicable Payment Dates or Period End Dates, as the case may be, of that party prior to the Termination Date will be the last day that is a Business Day in the month that is the specified number of months after the month in which the preceding applicable Payment Date or Period End Date occurred.