折に|おりに

On the occasion of; when (a chance arises). A formal, courteous way to refer to a particular occasion, such as a visit, at which something is done or hoped to be done.

A formal expression typical of letters, speeches, business correspondence and polite conversation. It is most often used for occasions such as visits and meetings, frequently to express a hope or intention to do something then. Compared with 際に or に際して, it sounds softer and more courteous, and it is less tied to procedures, notices or major undertakings.

• に際してN2「に際して」 is used specifically at the start of important events such as meetings or graduations, and the following clause is typically a speech, an expression of thanks, or procedures; it attaches only to nouns or verbs in dictionary form, never the た form. It carries no nuance of 'seizing an opportunity,' so invitations like 「お越しの折には」 cannot be rewritten with it.
• 際にN3The two attach in the same way, and both are more formal than 「時に」. 「際に」 leans toward official/business use, appearing often in notices and procedural explanations, while 「折に」 is more elegant, carries a nuance of 'opportunity,' and is common in letters and speeches; in polite set phrases such as 「ご来日の折／際に」 the two can often be interchanged.
• にあたってN2「にあたって」 is used when facing an important occasion to give a speech, state a position, or explain the preparations being made, and the verb attaches only in dictionary form. 「折に」 broadly refers to some occasion or opportunity, can take the た form to describe what happened at the time, and is often used for invitations, so the two are generally not interchangeable.
• にあたりN2「にあたり」 mostly marks the occasion of arriving at or embarking on something, and is followed by speeches, preparations, explanations, and the like; it attaches directly to nouns without 「の」 and does not take the た form. To describe something that happened to occur at that time, or to invite someone to visit 'when you have a chance,' use 「折に」.
• たびにN3「たびに」 means 'every time..., always...' and requires the preceding action to be repeated; 「折に」 points to a particular occasion or moment, has no 'every time' meaning, and is more elegant and formal. 「たびに」 is neutral in register, does not attach to the た form, and the two are generally not interchangeable.
• 場合はN4「場合は」 sets up situations that may arise and is followed by countermeasures, rules, or outcomes, appearing mostly in notices and manuals; 「折に」 refers to an occasion or moment, has an elegant tone, and can describe a one-time past event, which 「場合は」 cannot. The two are generally not interchangeable.

際に and に際して also mean 'when/on the occasion of' but are more neutral or businesslike and suit procedures and notices, while にあたって and にあたり stress preparing for a significant undertaking. たびに means 'every time' and 場合は marks a hypothetical case, whereas 折に refers courteously to a particular occasion, often seen as an opportunity.